id stringlengths 11 15 | title stringlengths 7 171 | problem stringlengths 9 4.33k | solution stringlengths 6 19k | problem_wikitext stringlengths 9 4.42k | solution_wikitext stringlengths 7 19.1k | proof_title stringlengths 9 171 | theorem_url stringlengths 34 198 | proof_url stringlengths 36 198 | categories listlengths 0 9 | theorem_references listlengths 0 36 | proof_references listlengths 0 253 |
|---|---|---|---|---|---|---|---|---|---|---|---|
proofwiki-9000 | Primitive of Power | Let $n \in \R: n \ne -1$.
Then:
:$\ds \int x^n \rd x = \frac {x^{n + 1} } {n + 1} + C$
where $C$ is an arbitrary constant. | Recall:
{{begin-eqn}}
{{eqn | n = 1
| l = \int x^m \arcsin x \rd x
| r = \frac {x^{m + 1} } {m + 1} \arcsin x - \frac 1 {m + 1} \int \frac {x^{m + 1} \rd x} {\sqrt {1 - x^2} }
| c = Primitive of $x^m \arcsin x$
}}
{{end-eqn}}
Then:
{{begin-eqn}}
{{eqn | l = \int x^m \arcsin \frac x a \rd x
| r =... | Let $n \in \R: n \ne -1$.
Then:
:$\ds \int x^n \rd x = \frac {x^{n + 1} } {n + 1} + C$
where $C$ is an [[Definition:Arbitrary Constant (Calculus)|arbitrary constant]]. | Recall:
{{begin-eqn}}
{{eqn | n = 1
| l = \int x^m \arcsin x \rd x
| r = \frac {x^{m + 1} } {m + 1} \arcsin x - \frac 1 {m + 1} \int \frac {x^{m + 1} \rd x} {\sqrt {1 - x^2} }
| c = [[Primitive of Power of x by Arcsine of x|Primitive of $x^m \arcsin x$]]
}}
{{end-eqn}}
Then:
{{begin-eqn}}
{{eqn | l... | Primitive of Power of x by Arcsine of x over a/Proof 1 | https://proofwiki.org/wiki/Primitive_of_Power | https://proofwiki.org/wiki/Primitive_of_Power_of_x_by_Arcsine_of_x_over_a/Proof_1 | [
"Primitive of Power",
"Primitives"
] | [
"Definition:Primitive (Calculus)/Constant of Integration"
] | [
"Primitive of Power of x by Arcsine of x",
"Primitive of Constant Multiple of Function",
"Primitive of Function of Constant Multiple"
] |
proofwiki-9001 | Primitive of Power | Let $n \in \R: n \ne -1$.
Then:
:$\ds \int x^n \rd x = \frac {x^{n + 1} } {n + 1} + C$
where $C$ is an arbitrary constant. | With a view to expressing the primitive in the form:
:$\ds \int u \frac {\d v} {\d x} \rd x = u v - \int v \frac {\d u} {\d x} \rd x$
let:
{{begin-eqn}}
{{eqn | l = u
| r = \arcsin \frac x a
| c =
}}
{{eqn | ll= \leadsto
| l = \frac {\d u} {\d x}
| r = \frac 1 {\sqrt {a^2 - x^2} }
| c = D... | Let $n \in \R: n \ne -1$.
Then:
:$\ds \int x^n \rd x = \frac {x^{n + 1} } {n + 1} + C$
where $C$ is an [[Definition:Arbitrary Constant (Calculus)|arbitrary constant]]. | With a view to expressing the [[Definition:Primitive|primitive]] in the form:
:$\ds \int u \frac {\d v} {\d x} \rd x = u v - \int v \frac {\d u} {\d x} \rd x$
let:
{{begin-eqn}}
{{eqn | l = u
| r = \arcsin \frac x a
| c =
}}
{{eqn | ll= \leadsto
| l = \frac {\d u} {\d x}
| r = \frac 1 {\sqrt {... | Primitive of Power of x by Arcsine of x over a/Proof 2 | https://proofwiki.org/wiki/Primitive_of_Power | https://proofwiki.org/wiki/Primitive_of_Power_of_x_by_Arcsine_of_x_over_a/Proof_2 | [
"Primitive of Power",
"Primitives"
] | [
"Definition:Primitive (Calculus)/Constant of Integration"
] | [
"Definition:Primitive",
"Derivative of Arcsine Function/Corollary",
"Primitive of Power",
"Integration by Parts",
"Primitive of Constant Multiple of Function"
] |
proofwiki-9002 | Primitive of Power | Let $n \in \R: n \ne -1$.
Then:
:$\ds \int x^n \rd x = \frac {x^{n + 1} } {n + 1} + C$
where $C$ is an arbitrary constant. | Recall:
{{begin-eqn}}
{{eqn | n = 1
| l = \int x^m \arctan x \rd x
| r = \frac {x^{m + 1} } {m + 1} \arctan x - \frac 1 {m + 1} \int \frac {x^{m + 1} \rd x} {x^2 + 1}
| c = Primitive of $x^m \arctan x$
}}
{{end-eqn}}
Then:
{{begin-eqn}}
{{eqn | l = \int x^m \arctan \frac x a \rd x
| r = \int a^m... | Let $n \in \R: n \ne -1$.
Then:
:$\ds \int x^n \rd x = \frac {x^{n + 1} } {n + 1} + C$
where $C$ is an [[Definition:Arbitrary Constant (Calculus)|arbitrary constant]]. | Recall:
{{begin-eqn}}
{{eqn | n = 1
| l = \int x^m \arctan x \rd x
| r = \frac {x^{m + 1} } {m + 1} \arctan x - \frac 1 {m + 1} \int \frac {x^{m + 1} \rd x} {x^2 + 1}
| c = [[Primitive of Power of x by Arctangent of x|Primitive of $x^m \arctan x$]]
}}
{{end-eqn}}
Then:
{{begin-eqn}}
{{eqn | l = \in... | Primitive of Power of x by Arctangent of x over a/Proof 1 | https://proofwiki.org/wiki/Primitive_of_Power | https://proofwiki.org/wiki/Primitive_of_Power_of_x_by_Arctangent_of_x_over_a/Proof_1 | [
"Primitive of Power",
"Primitives"
] | [
"Definition:Primitive (Calculus)/Constant of Integration"
] | [
"Primitive of Power of x by Arctangent of x",
"Primitive of Constant Multiple of Function",
"Primitive of Function of Constant Multiple"
] |
proofwiki-9003 | Primitive of Power | Let $n \in \R: n \ne -1$.
Then:
:$\ds \int x^n \rd x = \frac {x^{n + 1} } {n + 1} + C$
where $C$ is an arbitrary constant. | With a view to expressing the primitive in the form:
:$\ds \int u \frac {\d v} {\d x} \rd x = u v - \int v \frac {\d u} {\d x} \rd x$
let:
{{begin-eqn}}
{{eqn | l = u
| r = \arctan \frac x a
| c =
}}
{{eqn | ll= \leadsto
| l = \frac {\d u} {\d x}
| r = \frac a {x^2 + a^2}
| c = Derivative... | Let $n \in \R: n \ne -1$.
Then:
:$\ds \int x^n \rd x = \frac {x^{n + 1} } {n + 1} + C$
where $C$ is an [[Definition:Arbitrary Constant (Calculus)|arbitrary constant]]. | With a view to expressing the [[Definition:Primitive|primitive]] in the form:
:$\ds \int u \frac {\d v} {\d x} \rd x = u v - \int v \frac {\d u} {\d x} \rd x$
let:
{{begin-eqn}}
{{eqn | l = u
| r = \arctan \frac x a
| c =
}}
{{eqn | ll= \leadsto
| l = \frac {\d u} {\d x}
| r = \frac a {x^2 + a... | Primitive of Power of x by Arctangent of x over a/Proof 2 | https://proofwiki.org/wiki/Primitive_of_Power | https://proofwiki.org/wiki/Primitive_of_Power_of_x_by_Arctangent_of_x_over_a/Proof_2 | [
"Primitive of Power",
"Primitives"
] | [
"Definition:Primitive (Calculus)/Constant of Integration"
] | [
"Definition:Primitive",
"Derivative of Arctangent Function/Corollary",
"Primitive of Power",
"Integration by Parts",
"Primitive of Constant Multiple of Function"
] |
proofwiki-9004 | Primitive of Power | Let $n \in \R: n \ne -1$.
Then:
:$\ds \int x^n \rd x = \frac {x^{n + 1} } {n + 1} + C$
where $C$ is an arbitrary constant. | Let $s \in \Z$.
{{begin-eqn}}
{{eqn | l = v
| r = x^s
| c =
}}
{{eqn | n = 1
| ll= \leadsto
| l = \frac {\d v} {\d x}
| r = s x^{s - 1}
| c = Power Rule for Derivatives
}}
{{end-eqn}}
Let $u \dfrac {\d v} {\d x} = x^m \paren {a x + b}^n$.
Then:
{{begin-eqn}}
{{eqn | l = u
| r... | Let $n \in \R: n \ne -1$.
Then:
:$\ds \int x^n \rd x = \frac {x^{n + 1} } {n + 1} + C$
where $C$ is an [[Definition:Arbitrary Constant (Calculus)|arbitrary constant]]. | Let $s \in \Z$.
{{begin-eqn}}
{{eqn | l = v
| r = x^s
| c =
}}
{{eqn | n = 1
| ll= \leadsto
| l = \frac {\d v} {\d x}
| r = s x^{s - 1}
| c = [[Power Rule for Derivatives]]
}}
{{end-eqn}}
Let $u \dfrac {\d v} {\d x} = x^m \paren {a x + b}^n$.
Then:
{{begin-eqn}}
{{eqn | l = u
... | Primitive of Power of x by Power of a x + b/Decrement of Power of a x + b/Proof 1 | https://proofwiki.org/wiki/Primitive_of_Power | https://proofwiki.org/wiki/Primitive_of_Power_of_x_by_Power_of_a_x_+_b/Decrement_of_Power_of_a_x_+_b/Proof_1 | [
"Primitive of Power",
"Primitives"
] | [
"Definition:Primitive (Calculus)/Constant of Integration"
] | [
"Power Rule for Derivatives",
"Primitive of Power of a x + b",
"Product Rule for Derivatives",
"Integration by Parts"
] |
proofwiki-9005 | Primitive of Power | Let $n \in \R: n \ne -1$.
Then:
:$\ds \int x^n \rd x = \frac {x^{n + 1} } {n + 1} + C$
where $C$ is an arbitrary constant. | From Primitive of Power of $a x + b$ by Power of $p x + q$: Decrement of Power:
:$\ds \int \paren {a x + b}^m \paren {p x + q}^n \rd x = \frac {\paren {a x + b}^{m + 1} \paren {p x + q}^n} {\paren {m + n + 1} a} - \frac {n \paren {b p - a q} } {\paren {m + n + 1} a} \int \paren {a x + b}^m \paren {p x + q}^{n - 1} \rd ... | Let $n \in \R: n \ne -1$.
Then:
:$\ds \int x^n \rd x = \frac {x^{n + 1} } {n + 1} + C$
where $C$ is an [[Definition:Arbitrary Constant (Calculus)|arbitrary constant]]. | From [[Primitive of Power of a x + b by Power of p x + q/Decrement of Power|Primitive of Power of $a x + b$ by Power of $p x + q$: Decrement of Power]]:
:$\ds \int \paren {a x + b}^m \paren {p x + q}^n \rd x = \frac {\paren {a x + b}^{m + 1} \paren {p x + q}^n} {\paren {m + n + 1} a} - \frac {n \paren {b p - a q} } {\p... | Primitive of Power of x by Power of a x + b/Decrement of Power of a x + b/Proof 2 | https://proofwiki.org/wiki/Primitive_of_Power | https://proofwiki.org/wiki/Primitive_of_Power_of_x_by_Power_of_a_x_+_b/Decrement_of_Power_of_a_x_+_b/Proof_2 | [
"Primitive of Power",
"Primitives"
] | [
"Definition:Primitive (Calculus)/Constant of Integration"
] | [
"Primitive of Power of a x + b by Power of p x + q/Decrement of Power"
] |
proofwiki-9006 | Primitive of Power | Let $n \in \R: n \ne -1$.
Then:
:$\ds \int x^n \rd x = \frac {x^{n + 1} } {n + 1} + C$
where $C$ is an arbitrary constant. | Let $s \in \Z$.
{{begin-eqn}}
{{eqn | l = v
| r = \paren {a x + b}^s
| c =
}}
{{eqn | n = 1
| ll= \leadsto
| l = \frac {\d v} {\d x}
| r = a s \paren {a x + b}^{s - 1}
| c = Power Rule for Derivatives and Derivatives of Function of $a x + b$
}}
{{end-eqn}}
Let $u \dfrac {\d v} {\d ... | Let $n \in \R: n \ne -1$.
Then:
:$\ds \int x^n \rd x = \frac {x^{n + 1} } {n + 1} + C$
where $C$ is an [[Definition:Arbitrary Constant (Calculus)|arbitrary constant]]. | Let $s \in \Z$.
{{begin-eqn}}
{{eqn | l = v
| r = \paren {a x + b}^s
| c =
}}
{{eqn | n = 1
| ll= \leadsto
| l = \frac {\d v} {\d x}
| r = a s \paren {a x + b}^{s - 1}
| c = [[Power Rule for Derivatives]] and [[Derivatives of Function of a x + b|Derivatives of Function of $a x + b... | Primitive of Power of x by Power of a x + b/Decrement of Power of x/Proof 1 | https://proofwiki.org/wiki/Primitive_of_Power | https://proofwiki.org/wiki/Primitive_of_Power_of_x_by_Power_of_a_x_+_b/Decrement_of_Power_of_x/Proof_1 | [
"Primitive of Power",
"Primitives"
] | [
"Definition:Primitive (Calculus)/Constant of Integration"
] | [
"Power Rule for Derivatives",
"Derivatives of Function of a x + b",
"Product Rule for Derivatives",
"Integration by Parts"
] |
proofwiki-9007 | Primitive of Power | Let $n \in \R: n \ne -1$.
Then:
:$\ds \int x^n \rd x = \frac {x^{n + 1} } {n + 1} + C$
where $C$ is an arbitrary constant. | From Primitive of Power of $a x + b$ by Power of $p x + q$: Decrement of Power:
:$\ds \int \paren {a x + b}^m \paren {p x + q}^n \rd x = \frac {\paren {a x + b}^{m + 1} \paren {p x + q}^n} {\paren {m + n + 1} a} - \frac {n \paren {b p - a q} } {\paren {m + n + 1} a} \int \paren {a x + b}^m \paren {p x + q}^{n - 1} \rd ... | Let $n \in \R: n \ne -1$.
Then:
:$\ds \int x^n \rd x = \frac {x^{n + 1} } {n + 1} + C$
where $C$ is an [[Definition:Arbitrary Constant (Calculus)|arbitrary constant]]. | From [[Primitive of Power of a x + b by Power of p x + q/Decrement of Power|Primitive of Power of $a x + b$ by Power of $p x + q$: Decrement of Power]]:
:$\ds \int \paren {a x + b}^m \paren {p x + q}^n \rd x = \frac {\paren {a x + b}^{m + 1} \paren {p x + q}^n} {\paren {m + n + 1} a} - \frac {n \paren {b p - a q} } {\p... | Primitive of Power of x by Power of a x + b/Decrement of Power of x/Proof 2 | https://proofwiki.org/wiki/Primitive_of_Power | https://proofwiki.org/wiki/Primitive_of_Power_of_x_by_Power_of_a_x_+_b/Decrement_of_Power_of_x/Proof_2 | [
"Primitive of Power",
"Primitives"
] | [
"Definition:Primitive (Calculus)/Constant of Integration"
] | [
"Primitive of Power of a x + b by Power of p x + q/Decrement of Power"
] |
proofwiki-9008 | Primitive of Power | Let $n \in \R: n \ne -1$.
Then:
:$\ds \int x^n \rd x = \frac {x^{n + 1} } {n + 1} + C$
where $C$ is an arbitrary constant. | From Primitive of Power of $x$ by Power of $a x + b$: Decrement of Power of $x$:
:$\ds \int x^m \paren {a x + b}^n \rd x = \frac {x^{m + 1} \paren {a x + b}^n} {m + n + 1} + \frac {n b} {m + n + 1} \int x^m \paren {a x + b}^{n - 1} \rd x$
Substituting $n + 1$ for $n$:
{{begin-eqn}}
{{eqn | l = \int x^m \paren {a x + b}... | Let $n \in \R: n \ne -1$.
Then:
:$\ds \int x^n \rd x = \frac {x^{n + 1} } {n + 1} + C$
where $C$ is an [[Definition:Arbitrary Constant (Calculus)|arbitrary constant]]. | From [[Primitive of Power of x by Power of a x + b/Decrement of Power of x|Primitive of Power of $x$ by Power of $a x + b$: Decrement of Power of $x$]]:
:$\ds \int x^m \paren {a x + b}^n \rd x = \frac {x^{m + 1} \paren {a x + b}^n} {m + n + 1} + \frac {n b} {m + n + 1} \int x^m \paren {a x + b}^{n - 1} \rd x$
Substi... | Primitive of Power of x by Power of a x + b/Increment of Power of a x + b/Proof 1 | https://proofwiki.org/wiki/Primitive_of_Power | https://proofwiki.org/wiki/Primitive_of_Power_of_x_by_Power_of_a_x_+_b/Increment_of_Power_of_a_x_+_b/Proof_1 | [
"Primitive of Power",
"Primitives"
] | [
"Definition:Primitive (Calculus)/Constant of Integration"
] | [
"Primitive of Power of x by Power of a x + b/Decrement of Power of x"
] |
proofwiki-9009 | Primitive of Power | Let $n \in \R: n \ne -1$.
Then:
:$\ds \int x^n \rd x = \frac {x^{n + 1} } {n + 1} + C$
where $C$ is an arbitrary constant. | From Primitive of Power of $a x + b$ by Power of $p x + q$: Increment of Power:
:$\ds \int \paren {a x + b}^m \paren {p x + q}^n \rd x = \frac 1 {\paren {n + 1} \paren {b p - a q} } \paren {\paren {a x + b}^{m + 1} \paren {p x + q}^{n + 1} - a \paren {m + n + 2} \int \paren {a x + b}^m \paren {p x + q}^{n + 1} \rd x}$
... | Let $n \in \R: n \ne -1$.
Then:
:$\ds \int x^n \rd x = \frac {x^{n + 1} } {n + 1} + C$
where $C$ is an [[Definition:Arbitrary Constant (Calculus)|arbitrary constant]]. | From [[Primitive of Power of a x + b by Power of p x + q/Increment of Power|Primitive of Power of $a x + b$ by Power of $p x + q$: Increment of Power]]:
:$\ds \int \paren {a x + b}^m \paren {p x + q}^n \rd x = \frac 1 {\paren {n + 1} \paren {b p - a q} } \paren {\paren {a x + b}^{m + 1} \paren {p x + q}^{n + 1} - a \pa... | Primitive of Power of x by Power of a x + b/Increment of Power of a x + b/Proof 2 | https://proofwiki.org/wiki/Primitive_of_Power | https://proofwiki.org/wiki/Primitive_of_Power_of_x_by_Power_of_a_x_+_b/Increment_of_Power_of_a_x_+_b/Proof_2 | [
"Primitive of Power",
"Primitives"
] | [
"Definition:Primitive (Calculus)/Constant of Integration"
] | [
"Primitive of Power of a x + b by Power of p x + q/Increment of Power"
] |
proofwiki-9010 | Primitive of Power | Let $n \in \R: n \ne -1$.
Then:
:$\ds \int x^n \rd x = \frac {x^{n + 1} } {n + 1} + C$
where $C$ is an arbitrary constant. | From Primitive of Power of $x$ by Power of $a x + b$: Decrement of Power of $x$:
:$\ds \int x^m \paren {a x + b}^n \rd x = \frac {x^m \paren {a x + b}^{n + 1} } {\paren {m + n + 1} a} - \frac {m b} {\paren {m + n + 1} a} \int x^{m - 1} \paren {a x + b}^n \rd x$
Substituting $m + 1$ for $m$:
{{begin-eqn}}
{{eqn | l = \i... | Let $n \in \R: n \ne -1$.
Then:
:$\ds \int x^n \rd x = \frac {x^{n + 1} } {n + 1} + C$
where $C$ is an [[Definition:Arbitrary Constant (Calculus)|arbitrary constant]]. | From [[Primitive of Power of x by Power of a x + b/Decrement of Power of x|Primitive of Power of $x$ by Power of $a x + b$: Decrement of Power of $x$]]:
:$\ds \int x^m \paren {a x + b}^n \rd x = \frac {x^m \paren {a x + b}^{n + 1} } {\paren {m + n + 1} a} - \frac {m b} {\paren {m + n + 1} a} \int x^{m - 1} \paren {a x... | Primitive of Power of x by Power of a x + b/Increment of Power of x/Proof 1 | https://proofwiki.org/wiki/Primitive_of_Power | https://proofwiki.org/wiki/Primitive_of_Power_of_x_by_Power_of_a_x_+_b/Increment_of_Power_of_x/Proof_1 | [
"Primitive of Power",
"Primitives"
] | [
"Definition:Primitive (Calculus)/Constant of Integration"
] | [
"Primitive of Power of x by Power of a x + b/Decrement of Power of x"
] |
proofwiki-9011 | Primitive of Power | Let $n \in \R: n \ne -1$.
Then:
:$\ds \int x^n \rd x = \frac {x^{n + 1} } {n + 1} + C$
where $C$ is an arbitrary constant. | From Primitive of Power of $a x + b$ by Power of $p x + q$: Increment of Power:
:$\ds \int \paren {a x + b}^m \paren {p x + q}^n \rd x = \frac 1 {\paren {n + 1} \paren {b p - a q} } \paren {\paren {a x + b}^{m + 1} \paren {p x + q}^{n + 1} - a \paren {m + n + 2} \int \paren {a x + b}^m \paren {p x + q}^{n + 1} \rd x}$
... | Let $n \in \R: n \ne -1$.
Then:
:$\ds \int x^n \rd x = \frac {x^{n + 1} } {n + 1} + C$
where $C$ is an [[Definition:Arbitrary Constant (Calculus)|arbitrary constant]]. | From [[Primitive of Power of a x + b by Power of p x + q/Increment of Power|Primitive of Power of $a x + b$ by Power of $p x + q$: Increment of Power]]:
:$\ds \int \paren {a x + b}^m \paren {p x + q}^n \rd x = \frac 1 {\paren {n + 1} \paren {b p - a q} } \paren {\paren {a x + b}^{m + 1} \paren {p x + q}^{n + 1} - a \pa... | Primitive of Power of x by Power of a x + b/Increment of Power of x/Proof 2 | https://proofwiki.org/wiki/Primitive_of_Power | https://proofwiki.org/wiki/Primitive_of_Power_of_x_by_Power_of_a_x_+_b/Increment_of_Power_of_x/Proof_2 | [
"Primitive of Power",
"Primitives"
] | [
"Definition:Primitive (Calculus)/Constant of Integration"
] | [
"Primitive of Power of a x + b by Power of p x + q/Increment of Power"
] |
proofwiki-9012 | Primitive of Power | Let $n \in \R: n \ne -1$.
Then:
:$\ds \int x^n \rd x = \frac {x^{n + 1} } {n + 1} + C$
where $C$ is an arbitrary constant. | {{begin-eqn}}
{{eqn | l = \map {\frac \d {\d x} } {\dfrac {x^{n + 1} } {n + 1} }
| r = \paren {n + 1} \paren {\dfrac {x^{\paren {n + 1} - 1} } {n + 1} }
| c = Power Rule for Derivatives
}}
{{eqn | r = x^n
| c =
}}
{{eqn | ll= \leadsto
| l = \int x^n \rd x
| r = \frac {x^{n + 1} } {n + 1} ... | Let $n \in \R: n \ne -1$.
Then:
:$\ds \int x^n \rd x = \frac {x^{n + 1} } {n + 1} + C$
where $C$ is an [[Definition:Arbitrary Constant (Calculus)|arbitrary constant]]. | {{begin-eqn}}
{{eqn | l = \map {\frac \d {\d x} } {\dfrac {x^{n + 1} } {n + 1} }
| r = \paren {n + 1} \paren {\dfrac {x^{\paren {n + 1} - 1} } {n + 1} }
| c = [[Power Rule for Derivatives]]
}}
{{eqn | r = x^n
| c =
}}
{{eqn | ll= \leadsto
| l = \int x^n \rd x
| r = \frac {x^{n + 1} } {n +... | Primitive of Power/Proof | https://proofwiki.org/wiki/Primitive_of_Power | https://proofwiki.org/wiki/Primitive_of_Power/Proof | [
"Primitive of Power",
"Primitives"
] | [
"Definition:Primitive (Calculus)/Constant of Integration"
] | [
"Power Rule for Derivatives"
] |
proofwiki-9013 | Sine Inequality | :$\size {\sin x} \le \size x$
for all $x \in \R$. | Let $\map f x = x - \sin x$.
By Derivative of Sine Function:
:$\map {f'} x = 1 - \cos x$
From Real Cosine Function is Bounded we know $\cos x \le 1$ for all $x$.
Hence $\map f x \ge 0$ for all $x$.
From Derivative of Monotone Function, $\map f x$ is increasing.
By Sine of Zero is Zero, $\map f 0 = 0$.
It follows that ... | :$\size {\sin x} \le \size x$
for all $x \in \R$. | Let $\map f x = x - \sin x$.
By [[Derivative of Sine Function]]:
:$\map {f'} x = 1 - \cos x$
From [[Real Cosine Function is Bounded]] we know $\cos x \le 1$ for all $x$.
Hence $\map f x \ge 0$ for all $x$.
From [[Derivative of Monotone Function]], $\map f x$ is [[Definition:Increasing Real Function|increasing]].
B... | Sine Inequality/Proof 1 | https://proofwiki.org/wiki/Sine_Inequality | https://proofwiki.org/wiki/Sine_Inequality/Proof_1 | [
"Sine Inequality",
"Sine Function",
"Inequalities",
"Named Theorems"
] | [] | [
"Derivative of Sine Function",
"Real Cosine Function is Bounded",
"Derivative of Monotone Function",
"Definition:Increasing/Real Function",
"Sine of Zero is Zero",
"Derivative of Composite Function",
"Derivative of Sine Function",
"Double Angle Formulas/Sine",
"Derivative of Monotone Function",
"D... |
proofwiki-9014 | Sine Inequality | :$\size {\sin x} \le \size x$
for all $x \in \R$. | For $x = 0$, the inequality is trivial, as $\sin 0 = 0$.
If $\size x \ge 1$, the inequality also clear as:
{{begin-eqn}}
{{eqn | l = \size {\sin x}
| o = \le
| r = 1
| c = Real Sine Function is Bounded
}}
{{eqn | o = \le
| r = \size x
}}
{{end-eqn}}
Now, suppose $0 < \size x < 1$
Then on the one... | :$\size {\sin x} \le \size x$
for all $x \in \R$. | For $x = 0$, the inequality is trivial, as $\sin 0 = 0$.
If $\size x \ge 1$, the inequality also clear as:
{{begin-eqn}}
{{eqn | l = \size {\sin x}
| o = \le
| r = 1
| c = [[Real Sine Function is Bounded]]
}}
{{eqn | o = \le
| r = \size x
}}
{{end-eqn}}
Now, suppose $0 < \size x < 1$
Then on... | Sine Inequality/Proof 2 | https://proofwiki.org/wiki/Sine_Inequality | https://proofwiki.org/wiki/Sine_Inequality/Proof_2 | [
"Sine Inequality",
"Sine Function",
"Inequalities",
"Named Theorems"
] | [] | [
"Real Sine Function is Bounded",
"Sum of Infinite Geometric Sequence",
"Sum of Infinite Geometric Sequence"
] |
proofwiki-9015 | Jordan's Inequality | :300pxthumbright
:$\dfrac {2 x} \pi \le \sin x \le x$
for all $x$ in the interval $\closedint 0 {\dfrac \pi 2}$ | The {{RHS}} inequality is true by Sine Inequality.
The {{LHS}} inequality is true for $x = 0$ and $x = \dfrac \pi 2$, where we have equality.
Now consider $x \in \openint 0 {\dfrac \pi 2}$.
From Shape of Sine Function, $\sin x$ is concave on the interval $\closedint 0 \pi$.
Letting $x_1 = 0$, $x_2 = x$ and $x_3 = \dfra... | :[[File:JordansInequality.png|300px|thumb|right]]
:$\dfrac {2 x} \pi \le \sin x \le x$
for all $x$ in the [[Definition:Closed Real Interval|interval]] $\closedint 0 {\dfrac \pi 2}$ | The {{RHS}} inequality is true by [[Sine Inequality]].
The {{LHS}} inequality is true for $x = 0$ and $x = \dfrac \pi 2$, where we have equality.
Now consider $x \in \openint 0 {\dfrac \pi 2}$.
From [[Shape of Sine Function]], $\sin x$ is [[Definition:Concave Real Function|concave]] on the [[Definition:Closed Real I... | Jordan's Inequality | https://proofwiki.org/wiki/Jordan's_Inequality | https://proofwiki.org/wiki/Jordan's_Inequality | [
"Sine Function"
] | [
"File:JordansInequality.png",
"Definition:Real Interval/Closed"
] | [
"Sine Inequality",
"Shape of Sine Function",
"Definition:Concave Real Function",
"Definition:Real Interval/Closed",
"Definition:Concave Real Function/Definition 2"
] |
proofwiki-9016 | Cosine Inequality | :$1 - \dfrac {x^2} 2 \le \cos x$
for all $x \in \R$. | Let $\map f x = \cos x - \paren {1 - \dfrac {x^2} 2}$.
By Derivative of Cosine Function:
:$\map {f'} x = x - \sin x$
From Sine Inequality, we know $\sin x \le x$ for $x \ge 0$.
Hence $\map {f'} x \ge 0$ for $x \ge 0$.
From Derivative of Monotone Function, $\map f x$ is increasing for $x \ge 0$.
By Cosine of Zero is One... | :$1 - \dfrac {x^2} 2 \le \cos x$
for all $x \in \R$. | Let $\map f x = \cos x - \paren {1 - \dfrac {x^2} 2}$.
By [[Derivative of Cosine Function]]:
:$\map {f'} x = x - \sin x$
From [[Sine Inequality]], we know $\sin x \le x$ for $x \ge 0$.
Hence $\map {f'} x \ge 0$ for $x \ge 0$.
From [[Derivative of Monotone Function]], $\map f x$ is [[Definition:Increasing|increasing... | Cosine Inequality | https://proofwiki.org/wiki/Cosine_Inequality | https://proofwiki.org/wiki/Cosine_Inequality | [
"Cosine Function"
] | [] | [
"Derivative of Cosine Function",
"Sine Inequality",
"Derivative of Monotone Function",
"Definition:Increasing",
"Cosine of Zero is One",
"Cosine Function is Even",
"Definition:Even Function",
"Category:Cosine Function"
] |
proofwiki-9017 | Laplace Transform of Constant Mapping | Let $a \in \R$ be a real constant.
Let $f_a: \R \to \R$ or $\C$ be the constant mapping, defined as:
:$\forall t \in \R: \map {f_a} t = a$
Let $\laptrans {f_a}$ be the Laplace transform of $f_a$.
Then:
:$\laptrans {\map {f_a} t} = \dfrac a s$
for $\map \Re s > a$. | {{begin-eqn}}
{{eqn | l = \laptrans {\map {f_a} t}
| r = \laptrans a
| c = {{Defof|Constant Mapping}}
}}
{{eqn | r = a \, \laptrans 1
| c = Linear Combination of Laplace Transforms
}}
{{eqn | r = a \frac 1 s
| c = Laplace Transform of 1
}}
{{eqn | r = \frac a s
}}
{{end-eqn}}
{{qed}} | Let $a \in \R$ be a [[Definition:Real Number|real]] [[Definition:Constant|constant]].
Let $f_a: \R \to \R$ or $\C$ be the [[Definition:Constant Mapping|constant mapping]], defined as:
:$\forall t \in \R: \map {f_a} t = a$
Let $\laptrans {f_a}$ be the [[Definition:Laplace Transform|Laplace transform]] of $f_a$.
The... | {{begin-eqn}}
{{eqn | l = \laptrans {\map {f_a} t}
| r = \laptrans a
| c = {{Defof|Constant Mapping}}
}}
{{eqn | r = a \, \laptrans 1
| c = [[Linear Combination of Laplace Transforms]]
}}
{{eqn | r = a \frac 1 s
| c = [[Laplace Transform of 1]]
}}
{{eqn | r = \frac a s
}}
{{end-eqn}}
{{qed}} | Laplace Transform of Constant Mapping/Proof 1 | https://proofwiki.org/wiki/Laplace_Transform_of_Constant_Mapping | https://proofwiki.org/wiki/Laplace_Transform_of_Constant_Mapping/Proof_1 | [
"Laplace Transform of Constant Mapping",
"Examples of Laplace Transforms",
"Constant Mappings"
] | [
"Definition:Real Number",
"Definition:Constant",
"Definition:Constant Mapping",
"Definition:Laplace Transform"
] | [
"Linear Combination of Laplace Transforms",
"Laplace Transform of 1"
] |
proofwiki-9018 | Laplace Transform of Constant Mapping | Let $a \in \R$ be a real constant.
Let $f_a: \R \to \R$ or $\C$ be the constant mapping, defined as:
:$\forall t \in \R: \map {f_a} t = a$
Let $\laptrans {f_a}$ be the Laplace transform of $f_a$.
Then:
:$\laptrans {\map {f_a} t} = \dfrac a s$
for $\map \Re s > a$. | {{begin-eqn}}
{{eqn | l = \laptrans {\map {f_a} t}
| r = \laptrans a
| c = {{Defof|Constant Mapping}}
}}
{{eqn | r = a \, \laptrans 1
| c = Linear Combination of Laplace Transforms
}}
{{eqn | r = a \int_0^{\to +\infty} e^{-s t} \rd t
| c = {{Defof|Laplace Transform}}
}}
{{eqn | r = \intlimits {-... | Let $a \in \R$ be a [[Definition:Real Number|real]] [[Definition:Constant|constant]].
Let $f_a: \R \to \R$ or $\C$ be the [[Definition:Constant Mapping|constant mapping]], defined as:
:$\forall t \in \R: \map {f_a} t = a$
Let $\laptrans {f_a}$ be the [[Definition:Laplace Transform|Laplace transform]] of $f_a$.
The... | {{begin-eqn}}
{{eqn | l = \laptrans {\map {f_a} t}
| r = \laptrans a
| c = {{Defof|Constant Mapping}}
}}
{{eqn | r = a \, \laptrans 1
| c = [[Linear Combination of Laplace Transforms]]
}}
{{eqn | r = a \int_0^{\to +\infty} e^{-s t} \rd t
| c = {{Defof|Laplace Transform}}
}}
{{eqn | r = \intlimit... | Laplace Transform of Constant Mapping/Proof 2 | https://proofwiki.org/wiki/Laplace_Transform_of_Constant_Mapping | https://proofwiki.org/wiki/Laplace_Transform_of_Constant_Mapping/Proof_2 | [
"Laplace Transform of Constant Mapping",
"Examples of Laplace Transforms",
"Constant Mappings"
] | [
"Definition:Real Number",
"Definition:Constant",
"Definition:Constant Mapping",
"Definition:Laplace Transform"
] | [
"Linear Combination of Laplace Transforms",
"Complex Exponential Tends to Zero",
"Exponential of Zero"
] |
proofwiki-9019 | Laplace Transform of Identity Mapping | Let $\laptrans f$ denote the Laplace transform of a function $f$.
Let $\map {I_\R} t$ denote the identity mapping on $\R$ for $t > 0$.
Then:
:$\laptrans {\map {I_\R} t} = \dfrac 1 {s^2}$
for $\map \Re s > 0$. | {{begin-eqn}}
{{eqn | l = \laptrans {\map {I_\R} t}
| r = \laptrans t
| c = {{Defof|Identity Mapping}}
}}
{{eqn | r = \int_0^{\to +\infty} t e^{-st} \rd t
| c = {{Defof|Laplace Transform}}
}}
{{end-eqn}}
From Integration by Parts:
:$\ds \int f g' \rd t = f g - \int f'g \rd t$
Here:
{{begin-eqn}}
{{eqn... | Let $\laptrans f$ denote the [[Definition:Laplace Transform|Laplace transform]] of a [[Definition:Function|function]] $f$.
Let $\map {I_\R} t$ denote the [[Definition:Identity Mapping|identity mapping]] on $\R$ for $t > 0$.
Then:
:$\laptrans {\map {I_\R} t} = \dfrac 1 {s^2}$
for $\map \Re s > 0$. | {{begin-eqn}}
{{eqn | l = \laptrans {\map {I_\R} t}
| r = \laptrans t
| c = {{Defof|Identity Mapping}}
}}
{{eqn | r = \int_0^{\to +\infty} t e^{-st} \rd t
| c = {{Defof|Laplace Transform}}
}}
{{end-eqn}}
From [[Integration by Parts]]:
:$\ds \int f g' \rd t = f g - \int f'g \rd t$
Here:
{{begin-eq... | Laplace Transform of Identity Mapping/Proof 2 | https://proofwiki.org/wiki/Laplace_Transform_of_Identity_Mapping | https://proofwiki.org/wiki/Laplace_Transform_of_Identity_Mapping/Proof_2 | [
"Laplace Transform of Identity Mapping",
"Examples of Laplace Transforms",
"Identity Mappings"
] | [
"Definition:Laplace Transform",
"Definition:Function",
"Definition:Identity Mapping"
] | [
"Integration by Parts",
"Derivative of Identity Function",
"Primitive of Exponential Function",
"Primitive of Exponential Function",
"Exponential of Zero and One",
"Exponent Combination Laws/Negative Power",
"Limit at Infinity of Polynomial over Complex Exponential"
] |
proofwiki-9020 | Laplace Transform of Positive Integer Power | Let $\laptrans f$ denote the Laplace transform of a function $f$.
Let $t^n: \R \to \R$ be $t$ to the $n$th power for some $n \in \N_{\ge 0}$.
Then:
:$\laptrans {t^n} = \dfrac {n!} {s^{n + 1} }$
for $\map \Re s > 0$. | {{begin-eqn}}
{{eqn | l = \laptrans {t^n}
| r = \int_0^{\to +\infty} t^n e^{-s t} \rd t
| c = {{Defof|Laplace Transform}}
}}
{{eqn | r = \intlimits {\frac {e^{-s t} } {-s} \sum_{k \mathop = 0}^n \paren {\paren {-1}^k \frac {n^{\underline k} t^{n - k} } {\paren {-s}^k} } } {t \mathop = 0} {t \mathop \to +\in... | Let $\laptrans f$ denote the [[Definition:Laplace Transform|Laplace transform]] of a [[Definition:Function|function]] $f$.
Let $t^n: \R \to \R$ be [[Definition:Integer Power|$t$ to the $n$th power]] for some $n \in \N_{\ge 0}$.
Then:
:$\laptrans {t^n} = \dfrac {n!} {s^{n + 1} }$
for $\map \Re s > 0$. | {{begin-eqn}}
{{eqn | l = \laptrans {t^n}
| r = \int_0^{\to +\infty} t^n e^{-s t} \rd t
| c = {{Defof|Laplace Transform}}
}}
{{eqn | r = \intlimits {\frac {e^{-s t} } {-s} \sum_{k \mathop = 0}^n \paren {\paren {-1}^k \frac {n^{\underline k} t^{n - k} } {\paren {-s}^k} } } {t \mathop = 0} {t \mathop \to +\in... | Laplace Transform of Positive Integer Power/Proof 1 | https://proofwiki.org/wiki/Laplace_Transform_of_Positive_Integer_Power | https://proofwiki.org/wiki/Laplace_Transform_of_Positive_Integer_Power/Proof_1 | [
"Laplace Transform of Positive Integer Power",
"Examples of Laplace Transforms"
] | [
"Definition:Laplace Transform",
"Definition:Function",
"Definition:Power (Algebra)/Integer"
] | [
"Primitive of Power of x by Exponential of a x",
"Exponential Tends to Zero and Infinity",
"Exponent Combination Laws/Negative Power",
"Integer to Power of Itself Falling is Factorial"
] |
proofwiki-9021 | Laplace Transform of Positive Integer Power | Let $\laptrans f$ denote the Laplace transform of a function $f$.
Let $t^n: \R \to \R$ be $t$ to the $n$th power for some $n \in \N_{\ge 0}$.
Then:
:$\laptrans {t^n} = \dfrac {n!} {s^{n + 1} }$
for $\map \Re s > 0$. | The proof proceeds by induction on $n$ for $t^n$.
For all $n \in \Z_{\ge 0}$, let $\map P n$ be the proposition:
:$\laptrans {t^n} = \dfrac {n!} { s^{n + 1} }$
=== Basis for the Induction ===
$\map P 0$ is the case:
{{begin-eqn}}
{{eqn | l = \laptrans {t^0}
| r = \laptrans 1
}}
{{eqn | r = \frac 1 s
| c = L... | Let $\laptrans f$ denote the [[Definition:Laplace Transform|Laplace transform]] of a [[Definition:Function|function]] $f$.
Let $t^n: \R \to \R$ be [[Definition:Integer Power|$t$ to the $n$th power]] for some $n \in \N_{\ge 0}$.
Then:
:$\laptrans {t^n} = \dfrac {n!} {s^{n + 1} }$
for $\map \Re s > 0$. | The proof proceeds by [[Principle of Mathematical Induction|induction]] on $n$ for $t^n$.
For all $n \in \Z_{\ge 0}$, let $\map P n$ be the [[Definition:Proposition|proposition]]:
:$\laptrans {t^n} = \dfrac {n!} { s^{n + 1} }$
=== Basis for the Induction ===
$\map P 0$ is the case:
{{begin-eqn}}
{{eqn | l = \laptr... | Laplace Transform of Positive Integer Power/Proof 2 | https://proofwiki.org/wiki/Laplace_Transform_of_Positive_Integer_Power | https://proofwiki.org/wiki/Laplace_Transform_of_Positive_Integer_Power/Proof_2 | [
"Laplace Transform of Positive Integer Power",
"Examples of Laplace Transforms"
] | [
"Definition:Laplace Transform",
"Definition:Function",
"Definition:Power (Algebra)/Integer"
] | [
"Principle of Mathematical Induction",
"Definition:Proposition",
"Laplace Transform of 1",
"Definition:Basis for the Induction",
"Definition:Induction Hypothesis",
"Definition:Induction Step",
"Integration by Parts",
"Power Rule for Derivatives",
"Primitive of Exponential Function",
"Exponent Comb... |
proofwiki-9022 | Primitive of Function under its Derivative | Let $f$ be a real function which is integrable.
Then:
:$\ds \int \frac {\map {f'} x} {\map f x} \rd x = \ln \size {\map f x} + C$
where $C$ is an arbitrary constant. | By Integration by Substitution (with appropriate renaming of variables):
:$\ds \int \map g u \rd u = \int \map g {\map f x} \map {f'} x \rd x$
Then:
{{begin-eqn}}
{{eqn | l = \int \frac {\map {f'} x} {\map f x} \rd x
| r = \int \frac {\d u} u
| c = putting $\map g u := \dfrac 1 u$
}}
{{eqn | r = \ln \size u... | Let $f$ be a [[Definition:Real Function|real function]] which is [[Definition:Integrable Function|integrable]].
Then:
:$\ds \int \frac {\map {f'} x} {\map f x} \rd x = \ln \size {\map f x} + C$
where $C$ is an [[Definition:Arbitrary Constant (Calculus)|arbitrary constant]]. | By [[Integration by Substitution]] (with appropriate renaming of variables):
:$\ds \int \map g u \rd u = \int \map g {\map f x} \map {f'} x \rd x$
Then:
{{begin-eqn}}
{{eqn | l = \int \frac {\map {f'} x} {\map f x} \rd x
| r = \int \frac {\d u} u
| c = putting $\map g u := \dfrac 1 u$
}}
{{eqn | r = \ln \... | Primitive of Function under its Derivative | https://proofwiki.org/wiki/Primitive_of_Function_under_its_Derivative | https://proofwiki.org/wiki/Primitive_of_Function_under_its_Derivative | [
"Primitive of Function under its Derivative",
"Primitives"
] | [
"Definition:Real Function",
"Definition:Integrable Function",
"Definition:Primitive (Calculus)/Constant of Integration"
] | [
"Integration by Substitution",
"Primitive of Reciprocal"
] |
proofwiki-9023 | Primitive of Exponential Function/General Result | Let $a \in \R_{>0}$ be a constant such that $a \ne 1$.
Then:
:$\ds \int a^x \rd x = \frac {a^x} {\ln a} + C$
where $C$ is an arbitrary constant. | {{begin-eqn}}
{{eqn | l = \map {\dfrac \d {\d x} } {a^x}
| r = a^x \ln a
| c = Derivative of General Exponential Function
}}
{{eqn | ll= \leadsto
| l = \map {\dfrac \d {\d x} } {\dfrac {a^x} {\ln a} }
| r = a^x
| c = Derivative of Constant Multiple
}}
{{eqn | ll= \leadsto
| l = \int ... | Let $a \in \R_{>0}$ be a [[Definition:Constant|constant]] such that $a \ne 1$.
Then:
:$\ds \int a^x \rd x = \frac {a^x} {\ln a} + C$
where $C$ is an [[Definition:Arbitrary Constant (Calculus)|arbitrary constant]]. | {{begin-eqn}}
{{eqn | l = \map {\dfrac \d {\d x} } {a^x}
| r = a^x \ln a
| c = [[Derivative of General Exponential Function]]
}}
{{eqn | ll= \leadsto
| l = \map {\dfrac \d {\d x} } {\dfrac {a^x} {\ln a} }
| r = a^x
| c = [[Derivative of Constant Multiple]]
}}
{{eqn | ll= \leadsto
| l... | Primitive of Exponential Function/General Result/Proof 1 | https://proofwiki.org/wiki/Primitive_of_Exponential_Function/General_Result | https://proofwiki.org/wiki/Primitive_of_Exponential_Function/General_Result/Proof_1 | [
"Primitive of Exponential Function"
] | [
"Definition:Constant",
"Definition:Primitive (Calculus)/Constant of Integration"
] | [
"Derivative of General Exponential Function",
"Derivative of Constant Multiple"
] |
proofwiki-9024 | Primitive of Exponential Function/General Result | Let $a \in \R_{>0}$ be a constant such that $a \ne 1$.
Then:
:$\ds \int a^x \rd x = \frac {a^x} {\ln a} + C$
where $C$ is an arbitrary constant. | Let $u = x \ln a$.
{{begin-eqn}}
{{eqn | l = \int a^x \rd x
| r = \int \map \exp {x \ln a} \rd x
| c = {{Defof|Power to Real Number}}
}}
{{eqn | r = \frac 1 {\ln a} \int \map \exp u \rd u
| c = Primitive of Function of Constant Multiple
}}
{{eqn | r = \frac {\map \exp u} {\ln a} + C
| c = Primit... | Let $a \in \R_{>0}$ be a [[Definition:Constant|constant]] such that $a \ne 1$.
Then:
:$\ds \int a^x \rd x = \frac {a^x} {\ln a} + C$
where $C$ is an [[Definition:Arbitrary Constant (Calculus)|arbitrary constant]]. | Let $u = x \ln a$.
{{begin-eqn}}
{{eqn | l = \int a^x \rd x
| r = \int \map \exp {x \ln a} \rd x
| c = {{Defof|Power to Real Number}}
}}
{{eqn | r = \frac 1 {\ln a} \int \map \exp u \rd u
| c = [[Primitive of Function of Constant Multiple]]
}}
{{eqn | r = \frac {\map \exp u} {\ln a} + C
| c = [... | Primitive of Exponential Function/General Result/Proof 2 | https://proofwiki.org/wiki/Primitive_of_Exponential_Function/General_Result | https://proofwiki.org/wiki/Primitive_of_Exponential_Function/General_Result/Proof_2 | [
"Primitive of Exponential Function"
] | [
"Definition:Constant",
"Definition:Primitive (Calculus)/Constant of Integration"
] | [
"Primitive of Function of Constant Multiple",
"Primitive of Exponential Function"
] |
proofwiki-9025 | Primitive of Tangent Function/Secant Form | :$\ds \int \tan x \rd x = \ln \size {\sec x} + C$
where $\sec x$ is defined. | {{begin-eqn}}
{{eqn | l = \int \tan x \rd x
| r = -\ln \size {\cos x} + C
| c = Primitive of $\tan x$: Cosine Form
}}
{{eqn | r = \ln \size {\frac 1 {\cos x} } + C
| c = Logarithm of Reciprocal
}}
{{eqn | r = \ln \size {\sec x} + C
| c = Secant is Reciprocal of Cosine
}}
{{end-eqn}}
{{qed}} | :$\ds \int \tan x \rd x = \ln \size {\sec x} + C$
where $\sec x$ is defined. | {{begin-eqn}}
{{eqn | l = \int \tan x \rd x
| r = -\ln \size {\cos x} + C
| c = [[Primitive of Tangent Function/Cosine Form|Primitive of $\tan x$: Cosine Form]]
}}
{{eqn | r = \ln \size {\frac 1 {\cos x} } + C
| c = [[Logarithm of Reciprocal]]
}}
{{eqn | r = \ln \size {\sec x} + C
| c = [[Secant... | Primitive of Tangent Function/Secant Form/Proof | https://proofwiki.org/wiki/Primitive_of_Tangent_Function/Secant_Form | https://proofwiki.org/wiki/Primitive_of_Tangent_Function/Secant_Form/Proof | [
"Primitive of Tangent Function"
] | [] | [
"Primitive of Tangent Function/Cosine Form",
"Logarithm of Reciprocal",
"Secant is Reciprocal of Cosine"
] |
proofwiki-9026 | Exponential of x not less than 1+x | :$e^x \ge 1 + x$
for all $x \in \R$. | For $x > - 1$:
{{begin-eqn}}
{{eqn | l = e^x
| r = \lim_{n \mathop \to \infty} \left({1 + \frac x n}\right)^n
| c = {{Defof|Real Exponential Function}}
}}
{{eqn | o = \ge
| r = \lim_{n \mathop \to \infty} \left({1 + x}\right)
| c = Bernoulli's Inequality
}}
{{eqn | r = 1 + x
| c =
}}
{{en... | :$e^x \ge 1 + x$
for all $x \in \R$. | For $x > - 1$:
{{begin-eqn}}
{{eqn | l = e^x
| r = \lim_{n \mathop \to \infty} \left({1 + \frac x n}\right)^n
| c = {{Defof|Real Exponential Function}}
}}
{{eqn | o = \ge
| r = \lim_{n \mathop \to \infty} \left({1 + x}\right)
| c = [[Bernoulli's Inequality]]
}}
{{eqn | r = 1 + x
| c =
}}... | Exponential of x not less than 1+x | https://proofwiki.org/wiki/Exponential_of_x_not_less_than_1+x | https://proofwiki.org/wiki/Exponential_of_x_not_less_than_1+x | [
"Exponential Function",
"Inequalities"
] | [] | [
"Bernoulli's Inequality",
"Definition:Positive Real Function",
"Category:Exponential Function",
"Category:Inequalities"
] |
proofwiki-9027 | Primitive of Cosecant Function/Cosecant minus Cotangent Form | :$\ds \int \csc x \rd x = \ln \size {\csc x - \cot x} + C$
where $\csc x - \cot x \ne 0$. | {{begin-eqn}}
{{eqn | l = \int \csc x \rd x
| r = \ln \size {\tan \frac x 2} + C
| c = Primitive of $\csc x$: Tangent Form
}}
{{eqn | r = \ln \size {\frac {1 - \cos x} {\sin x} } + C
| c = {{Corollary|Half Angle Formula for Tangent|2}}
}}
{{eqn | r = \ln \size {\frac 1 {\sin x} - \frac {\cos x} {\sin ... | :$\ds \int \csc x \rd x = \ln \size {\csc x - \cot x} + C$
where $\csc x - \cot x \ne 0$. | {{begin-eqn}}
{{eqn | l = \int \csc x \rd x
| r = \ln \size {\tan \frac x 2} + C
| c = [[Primitive of Cosecant Function/Tangent Form|Primitive of $\csc x$: Tangent Form]]
}}
{{eqn | r = \ln \size {\frac {1 - \cos x} {\sin x} } + C
| c = {{Corollary|Half Angle Formula for Tangent|2}}
}}
{{eqn | r = \ln... | Primitive of Cosecant Function/Cosecant minus Cotangent Form/Proof | https://proofwiki.org/wiki/Primitive_of_Cosecant_Function/Cosecant_minus_Cotangent_Form | https://proofwiki.org/wiki/Primitive_of_Cosecant_Function/Cosecant_minus_Cotangent_Form/Proof | [
"Primitive of Cosecant Function"
] | [] | [
"Primitive of Cosecant Function/Tangent Form"
] |
proofwiki-9028 | Primitive of Cosecant Function/Tangent Form | :$\ds \int \csc x \rd x = \ln \size {\tan \frac x 2} + C$
where $\tan \dfrac x 2 \ne 0$. | {{Mistake|Review grammar in the below. Recall the rule about when capital letters are used. See Help:Editing/House Style#Capital Letters begin Sentences.}}
Since:
:$\tan \dfrac x 2 \ne 0$
It follows that $\forall n \in \Z$:
:$ 2\pi n < x < 2\pi n + \pi$
So:
:$\csc x + \cot x \ne 0$
And:
:$\dfrac {1 + \cos x} {\sin x} ... | :$\ds \int \csc x \rd x = \ln \size {\tan \frac x 2} + C$
where $\tan \dfrac x 2 \ne 0$. | {{Mistake|Review grammar in the below. Recall the rule about when capital letters are used. See [[Help:Editing/House Style#Capital Letters begin Sentences]].}}
Since:
:$\tan \dfrac x 2 \ne 0$
It follows that $\forall n \in \Z$:
:$ 2\pi n < x < 2\pi n + \pi$
So:
:$\csc x + \cot x \ne 0$
And:
:$\dfrac {1 + \cos x} {\... | Primitive of Cosecant Function/Tangent Form/Proof 1 | https://proofwiki.org/wiki/Primitive_of_Cosecant_Function/Tangent_Form | https://proofwiki.org/wiki/Primitive_of_Cosecant_Function/Tangent_Form/Proof_1 | [
"Primitive of Cosecant Function"
] | [] | [
"Help:Editing/House Style",
"Primitive of Cosecant Function/Cosecant plus Cotangent Form",
"Logarithm of Reciprocal"
] |
proofwiki-9029 | Primitive of Cosecant Function/Tangent Form | :$\ds \int \csc x \rd x = \ln \size {\tan \frac x 2} + C$
where $\tan \dfrac x 2 \ne 0$. | {{begin-eqn}}
{{eqn | l = \int \csc x \rd x
| r = \int \frac 1 {\sin x} \rd x
| c = Cosecant is Reciprocal of Sine
}}
{{end-eqn}}
We make the Weierstrass Substitution:
{{begin-eqn}}
{{eqn | l = u
| r = \tan \frac x 2
}}
{{eqn | ll= \leadsto
| l = \sin x
| r = \frac {2 u} {u^2 + 1}
}}
{{eq... | :$\ds \int \csc x \rd x = \ln \size {\tan \frac x 2} + C$
where $\tan \dfrac x 2 \ne 0$. | {{begin-eqn}}
{{eqn | l = \int \csc x \rd x
| r = \int \frac 1 {\sin x} \rd x
| c = [[Cosecant is Reciprocal of Sine]]
}}
{{end-eqn}}
We make the [[Weierstrass Substitution]]:
{{begin-eqn}}
{{eqn | l = u
| r = \tan \frac x 2
}}
{{eqn | ll= \leadsto
| l = \sin x
| r = \frac {2 u} {u^2 +... | Primitive of Cosecant Function/Tangent Form/Proof 2 | https://proofwiki.org/wiki/Primitive_of_Cosecant_Function/Tangent_Form | https://proofwiki.org/wiki/Primitive_of_Cosecant_Function/Tangent_Form/Proof_2 | [
"Primitive of Cosecant Function"
] | [] | [
"Cosecant is Reciprocal of Sine",
"Weierstrass Substitution",
"Primitive of Reciprocal"
] |
proofwiki-9030 | Primitive of Square of Secant Function | :$\ds \int \sec^2 x \rd x = \tan x + C$
where $C$ is an arbitrary constant. | From Derivative of Tangent Function:
:$\map {\dfrac \d {\d x} } {\tan x} = \sec^2 x$
The result follows from the definition of primitive.
{{Qed}} | :$\ds \int \sec^2 x \rd x = \tan x + C$
where $C$ is an [[Definition:Arbitrary Constant (Calculus)|arbitrary constant]]. | From [[Derivative of Tangent Function]]:
:$\map {\dfrac \d {\d x} } {\tan x} = \sec^2 x$
The result follows from the definition of [[Definition:Primitive (Calculus)|primitive]].
{{Qed}} | Primitive of Square of Secant Function | https://proofwiki.org/wiki/Primitive_of_Square_of_Secant_Function | https://proofwiki.org/wiki/Primitive_of_Square_of_Secant_Function | [
"Primitives involving Secant Function",
"Tangent Function"
] | [
"Definition:Primitive (Calculus)/Constant of Integration"
] | [
"Derivative of Tangent Function",
"Definition:Primitive (Calculus)"
] |
proofwiki-9031 | Primitive of Square of Cosecant Function | :$\ds \int \csc^2 x \rd x = -\cot x + C$ | From Derivative of Cotangent Function:
:$\dfrac \d {\d x} \cot x = -\csc^2 x$
The result follows from the definition of primitive.
{{Qed}} | :$\ds \int \csc^2 x \rd x = -\cot x + C$ | From [[Derivative of Cotangent Function]]:
:$\dfrac \d {\d x} \cot x = -\csc^2 x$
The result follows from the definition of [[Definition:Primitive (Calculus)|primitive]].
{{Qed}} | Primitive of Square of Cosecant Function | https://proofwiki.org/wiki/Primitive_of_Square_of_Cosecant_Function | https://proofwiki.org/wiki/Primitive_of_Square_of_Cosecant_Function | [
"Primitives involving Cosecant Function",
"Cotangent Function"
] | [] | [
"Derivative of Cotangent Function",
"Definition:Primitive (Calculus)"
] |
proofwiki-9032 | Primitive of Square of Tangent Function | :$\ds \int \tan^2 x \rd x = \tan x - x + C$ | {{begin-eqn}}
{{eqn | l = \int \tan^2 x \rd x
| r = \int \paren {\sec^2 x - 1} \rd x
| c = Difference of Squares of Secant and Tangent
}}
{{eqn | r = \int \sec^2 x \rd x + \int \paren {-1} \rd x
| c = Linear Combination of Primitives
}}
{{eqn | r = \tan x + C + \int \paren {-1} \rd x
| c = Primi... | :$\ds \int \tan^2 x \rd x = \tan x - x + C$ | {{begin-eqn}}
{{eqn | l = \int \tan^2 x \rd x
| r = \int \paren {\sec^2 x - 1} \rd x
| c = [[Difference of Squares of Secant and Tangent]]
}}
{{eqn | r = \int \sec^2 x \rd x + \int \paren {-1} \rd x
| c = [[Linear Combination of Primitives]]
}}
{{eqn | r = \tan x + C + \int \paren {-1} \rd x
| c... | Primitive of Square of Tangent Function/Proof 1 | https://proofwiki.org/wiki/Primitive_of_Square_of_Tangent_Function | https://proofwiki.org/wiki/Primitive_of_Square_of_Tangent_Function/Proof_1 | [
"Primitive of Square of Tangent Function",
"Primitives involving Tangent Function"
] | [] | [
"Sum of Squares of Sine and Cosine/Corollary 1",
"Linear Combination of Integrals/Indefinite",
"Primitive of Square of Secant Function",
"Primitive of Constant"
] |
proofwiki-9033 | Primitive of Square of Tangent Function | :$\ds \int \tan^2 x \rd x = \tan x - x + C$ | {{begin-eqn}}
{{eqn | l = I_n
| r = \int \tan^n x \rd x
}}
{{eqn | r = \frac {\tan^{n - 1} x} {n - 1} - I_{n - 2}
| c = Reduction Formula for Integral of Power of Tangent
}}
{{eqn | l = I_0
| r = \int \paren {\tan x}^0 \rd x
}}
{{eqn | r = \int \d x
| c =
}}
{{eqn | r = x + C
| c = Primitive of Constant
}}
{{eqn ... | :$\ds \int \tan^2 x \rd x = \tan x - x + C$ | {{begin-eqn}}
{{eqn | l = I_n
| r = \int \tan^n x \rd x
}}
{{eqn | r = \frac {\tan^{n - 1} x} {n - 1} - I_{n - 2}
| c = [[Reduction Formula for Integral of Power of Tangent]]
}}
{{eqn | l = I_0
| r = \int \paren {\tan x}^0 \rd x
}}
{{eqn | r = \int \d x
| c =
}}
{{eqn | r = x + C
| c = [[Primitive of Constant]]
}... | Primitive of Square of Tangent Function/Proof 2 | https://proofwiki.org/wiki/Primitive_of_Square_of_Tangent_Function | https://proofwiki.org/wiki/Primitive_of_Square_of_Tangent_Function/Proof_2 | [
"Primitive of Square of Tangent Function",
"Primitives involving Tangent Function"
] | [] | [
"Reduction Formula for Integral of Power of Tangent",
"Primitive of Constant"
] |
proofwiki-9034 | Primitive of Square of Cotangent Function | :$\ds \int \cot^2 x \rd x = -\cot x - x + C$
where $C$ is an arbitrary constant. | {{begin-eqn}}
{{eqn | l = \int \cot^2 x \rd x
| r = \int \paren {\csc^2 x - 1} \rd x
| c = Difference of Squares of Cosecant and Cotangent
}}
{{eqn | r = \int \csc^2 x \rd x + \int \paren {-1} \rd x
| c = Linear Combination of Primitives
}}
{{eqn | r = -\cot x + C + \int \paren {-1} \rd x
| c = ... | :$\ds \int \cot^2 x \rd x = -\cot x - x + C$
where $C$ is an [[Definition:Arbitrary Constant (Calculus)|arbitrary constant]]. | {{begin-eqn}}
{{eqn | l = \int \cot^2 x \rd x
| r = \int \paren {\csc^2 x - 1} \rd x
| c = [[Difference of Squares of Cosecant and Cotangent]]
}}
{{eqn | r = \int \csc^2 x \rd x + \int \paren {-1} \rd x
| c = [[Linear Combination of Primitives]]
}}
{{eqn | r = -\cot x + C + \int \paren {-1} \rd x
... | Primitive of Square of Cotangent Function | https://proofwiki.org/wiki/Primitive_of_Square_of_Cotangent_Function | https://proofwiki.org/wiki/Primitive_of_Square_of_Cotangent_Function | [
"Primitives involving Cotangent Function"
] | [
"Definition:Primitive (Calculus)/Constant of Integration"
] | [
"Sum of Squares of Sine and Cosine/Corollary 2",
"Linear Combination of Integrals/Indefinite",
"Primitive of Square of Cosecant Function",
"Primitive of Constant"
] |
proofwiki-9035 | Primitive of Square of Sine Function | :$\ds \int \sin^2 x \rd x = \frac x 2 - \frac {\sin 2 x} 4 + C$ | {{begin-eqn}}
{{eqn | l = \int \sin^2 x \rd x
| r = \int \paren {\frac {1 - \cos 2 x} 2} \rd x
| c = Square of Sine
}}
{{eqn | r = \frac 1 2 \int \d x - \frac 1 2 \int \cos 2 x \rd x
| c = Linear Combination of Primitives
}}
{{eqn | r = \frac x 2 - \frac 1 2 \int \cos 2 x \rd x + C
| c = Primiti... | :$\ds \int \sin^2 x \rd x = \frac x 2 - \frac {\sin 2 x} 4 + C$ | {{begin-eqn}}
{{eqn | l = \int \sin^2 x \rd x
| r = \int \paren {\frac {1 - \cos 2 x} 2} \rd x
| c = [[Square of Sine]]
}}
{{eqn | r = \frac 1 2 \int \d x - \frac 1 2 \int \cos 2 x \rd x
| c = [[Linear Combination of Primitives]]
}}
{{eqn | r = \frac x 2 - \frac 1 2 \int \cos 2 x \rd x + C
| c =... | Primitive of Square of Sine Function | https://proofwiki.org/wiki/Primitive_of_Square_of_Sine_Function | https://proofwiki.org/wiki/Primitive_of_Square_of_Sine_Function | [
"Primitive of Square of Sine Function",
"Primitives involving Sine Function"
] | [] | [
"Power Reduction Formulas/Sine Squared",
"Linear Combination of Integrals/Indefinite",
"Primitive of Constant",
"Primitive of Cosine Function/Corollary"
] |
proofwiki-9036 | Primitive of Square of Cosine Function | :$\ds \int \cos^2 x \rd x = \frac x 2 + \frac {\sin 2 x} 4 + C$ | {{begin-eqn}}
{{eqn | l = \int \cos^2 x \rd x
| r = \int \paren {\frac {1 + \cos 2 x} 2} \rd x
| c = Square of Cosine
}}
{{eqn | r = \int \frac 1 2 \rd x + \int \paren {\frac {\cos 2 x} 2} \rd x
| c = Linear Combination of Primitives
}}
{{eqn | r = \frac x 2 + C + \int \paren {\frac {\cos 2 x} 2} \rd ... | :$\ds \int \cos^2 x \rd x = \frac x 2 + \frac {\sin 2 x} 4 + C$ | {{begin-eqn}}
{{eqn | l = \int \cos^2 x \rd x
| r = \int \paren {\frac {1 + \cos 2 x} 2} \rd x
| c = [[Square of Cosine]]
}}
{{eqn | r = \int \frac 1 2 \rd x + \int \paren {\frac {\cos 2 x} 2} \rd x
| c = [[Linear Combination of Primitives]]
}}
{{eqn | r = \frac x 2 + C + \int \paren {\frac {\cos 2 x}... | Primitive of Square of Cosine Function/Proof 1 | https://proofwiki.org/wiki/Primitive_of_Square_of_Cosine_Function | https://proofwiki.org/wiki/Primitive_of_Square_of_Cosine_Function/Proof_1 | [
"Primitive of Square of Cosine Function",
"Primitives involving Cosine Function"
] | [] | [
"Power Reduction Formulas/Cosine Squared",
"Linear Combination of Integrals/Indefinite",
"Primitive of Constant",
"Primitive of Constant Multiple of Function",
"Primitive of Function of Constant Multiple",
"Primitive of Cosine Function",
"Primitive of Function of Constant Multiple",
"Primitive of Cosi... |
proofwiki-9037 | Primitive of Square of Cosine Function | :$\ds \int \cos^2 x \rd x = \frac x 2 + \frac {\sin 2 x} 4 + C$ | {{begin-eqn}}
{{eqn | l = I_n
| r = \int \cos^n x \rd x
}}
{{eqn | r = \dfrac {\cos^{n - 1} x \sin x} n + \dfrac {n - 1} n I_{n-2}
| c = Reduction Formula for Integral of Power of Cosine
}}
{{eqn | l = I_0
| r = \int \left({\cos x}\right)^0 \rd x
| c =
}}
{{eqn | r = \int \rd x
| c =
}}
{{eqn | r = x + C
| c = ... | :$\ds \int \cos^2 x \rd x = \frac x 2 + \frac {\sin 2 x} 4 + C$ | {{begin-eqn}}
{{eqn | l = I_n
| r = \int \cos^n x \rd x
}}
{{eqn | r = \dfrac {\cos^{n - 1} x \sin x} n + \dfrac {n - 1} n I_{n-2}
| c = [[Reduction Formula for Integral of Power of Cosine]]
}}
{{eqn | l = I_0
| r = \int \left({\cos x}\right)^0 \rd x
| c =
}}
{{eqn | r = \int \rd x
| c =
}}
{{eqn | r = x + C
| ... | Primitive of Square of Cosine Function/Proof 2 | https://proofwiki.org/wiki/Primitive_of_Square_of_Cosine_Function | https://proofwiki.org/wiki/Primitive_of_Square_of_Cosine_Function/Proof_2 | [
"Primitive of Square of Cosine Function",
"Primitives involving Cosine Function"
] | [] | [
"Reduction Formula for Integral of Power of Cosine",
"Primitive of Constant",
"Double Angle Formulas/Sine"
] |
proofwiki-9038 | Primitive of Square of Cosine Function | :$\ds \int \cos^2 x \rd x = \frac x 2 + \frac {\sin 2 x} 4 + C$ | {{begin-eqn}}
{{eqn | l = \int \cos^2 x \rd x
| r = \frac 1 4 \int \paren {e^{i x} + e^{-i x} }^2 \rd x
| c = Euler's Cosine Identity
}}
{{eqn | r = \frac 1 4 \int \paren {e^{2 i x} + 2 + e^{-2 i x} } \rd x
}}
{{eqn | r = \frac 1 4 \paren {\frac{e^{2 i x} - e^{-2 i x} } {2 i} + 2 x} + C
| c = Primiti... | :$\ds \int \cos^2 x \rd x = \frac x 2 + \frac {\sin 2 x} 4 + C$ | {{begin-eqn}}
{{eqn | l = \int \cos^2 x \rd x
| r = \frac 1 4 \int \paren {e^{i x} + e^{-i x} }^2 \rd x
| c = [[Euler's Cosine Identity]]
}}
{{eqn | r = \frac 1 4 \int \paren {e^{2 i x} + 2 + e^{-2 i x} } \rd x
}}
{{eqn | r = \frac 1 4 \paren {\frac{e^{2 i x} - e^{-2 i x} } {2 i} + 2 x} + C
| c = [[P... | Primitive of Square of Cosine Function/Proof 3 | https://proofwiki.org/wiki/Primitive_of_Square_of_Cosine_Function | https://proofwiki.org/wiki/Primitive_of_Square_of_Cosine_Function/Proof_3 | [
"Primitive of Square of Cosine Function",
"Primitives involving Cosine Function"
] | [] | [
"Euler's Cosine Identity",
"Primitive of Exponential of a x",
"Primitive of Constant",
"Euler's Sine Identity"
] |
proofwiki-9039 | Primitive of Product of Secant and Tangent | :$\ds \int \sec x \tan x \rd x = \sec x + C$
where $C$ is an arbitrary constant. | From Derivative of Secant Function:
:$\dfrac \d {\d x} \sec x = \sec x \tan x$
The result follows from the definition of primitive.
{{Qed}} | :$\ds \int \sec x \tan x \rd x = \sec x + C$
where $C$ is an [[Definition:Arbitrary Constant (Calculus)|arbitrary constant]]. | From [[Derivative of Secant Function]]:
:$\dfrac \d {\d x} \sec x = \sec x \tan x$
The result follows from the definition of [[Definition:Primitive (Calculus)|primitive]].
{{Qed}} | Primitive of Product of Secant and Tangent | https://proofwiki.org/wiki/Primitive_of_Product_of_Secant_and_Tangent | https://proofwiki.org/wiki/Primitive_of_Product_of_Secant_and_Tangent | [
"Primitives involving Tangent Function",
"Primitives involving Secant Function"
] | [
"Definition:Primitive (Calculus)/Constant of Integration"
] | [
"Derivative of Secant Function",
"Definition:Primitive (Calculus)"
] |
proofwiki-9040 | Primitive of Product of Cosecant and Cotangent | :$\ds \int \csc x \cot x \rd x = -\csc x + C$
where $C$ is an arbitrary constant. | From Derivative of Cosecant Function:
:$\dfrac \d {\d x} \csc x = -\csc x \cot x$
The result follows from the definition of primitive.
{{Qed}} | :$\ds \int \csc x \cot x \rd x = -\csc x + C$
where $C$ is an [[Definition:Arbitrary Constant (Calculus)|arbitrary constant]]. | From [[Derivative of Cosecant Function]]:
:$\dfrac \d {\d x} \csc x = -\csc x \cot x$
The result follows from the definition of [[Definition:Primitive (Calculus)|primitive]].
{{Qed}} | Primitive of Product of Cosecant and Cotangent | https://proofwiki.org/wiki/Primitive_of_Product_of_Cosecant_and_Cotangent | https://proofwiki.org/wiki/Primitive_of_Product_of_Cosecant_and_Cotangent | [
"Primitives involving Cosecant Function",
"Primitives involving Cotangent Function"
] | [
"Definition:Primitive (Calculus)/Constant of Integration"
] | [
"Derivative of Cosecant Function",
"Definition:Primitive (Calculus)"
] |
proofwiki-9041 | Primitive of Hyperbolic Sine Function | :$\ds \int \sinh x \rd x = \cosh x + C$ | From Derivative of Hyperbolic Cosine:
:$\map {\dfrac \d {\d x} } {\cosh x} = \sinh x$
The result follows from the definition of primitive.
{{qed}} | :$\ds \int \sinh x \rd x = \cosh x + C$ | From [[Derivative of Hyperbolic Cosine]]:
:$\map {\dfrac \d {\d x} } {\cosh x} = \sinh x$
The result follows from the definition of [[Definition:Primitive (Calculus)|primitive]].
{{qed}} | Primitive of Hyperbolic Sine Function | https://proofwiki.org/wiki/Primitive_of_Hyperbolic_Sine_Function | https://proofwiki.org/wiki/Primitive_of_Hyperbolic_Sine_Function | [
"Primitives of Hyperbolic Functions",
"Hyperbolic Sine Function",
"Primitives involving Hyperbolic Sine Function"
] | [] | [
"Derivative of Hyperbolic Cosine",
"Definition:Primitive (Calculus)"
] |
proofwiki-9042 | Primitive of Hyperbolic Cosine Function | :$\ds \int \cosh x \rd x = \sinh x + C$ | From Derivative of Hyperbolic Sine:
:$\map {\dfrac \d {\d x} } {\sinh x} = \cosh x$
The result follows from the definition of primitive.
{{qed}} | :$\ds \int \cosh x \rd x = \sinh x + C$ | From [[Derivative of Hyperbolic Sine]]:
:$\map {\dfrac \d {\d x} } {\sinh x} = \cosh x$
The result follows from the definition of [[Definition:Primitive (Calculus)|primitive]].
{{qed}} | Primitive of Hyperbolic Cosine Function | https://proofwiki.org/wiki/Primitive_of_Hyperbolic_Cosine_Function | https://proofwiki.org/wiki/Primitive_of_Hyperbolic_Cosine_Function | [
"Primitives of Hyperbolic Functions",
"Hyperbolic Cosine Function",
"Primitives involving Hyperbolic Cosine Function"
] | [] | [
"Derivative of Hyperbolic Sine",
"Definition:Primitive (Calculus)"
] |
proofwiki-9043 | Primitive of Hyperbolic Tangent Function | :$\ds \int \tanh x \rd x = \map \ln {\cosh x} + C$ | {{begin-eqn}}
{{eqn | l = \int \tanh x \rd x
| r = \int \frac {\sinh x} {\cosh x} \rd x
| c = {{Defof|Hyperbolic Tangent}}
}}
{{eqn | r = \int \frac {\paren {\cosh x}'} {\cosh x} \rd x
| c = Derivative of Hyperbolic Cosine
}}
{{eqn | r = \ln \size {\cosh x} + C
| c = Primitive of Function under ... | :$\ds \int \tanh x \rd x = \map \ln {\cosh x} + C$ | {{begin-eqn}}
{{eqn | l = \int \tanh x \rd x
| r = \int \frac {\sinh x} {\cosh x} \rd x
| c = {{Defof|Hyperbolic Tangent}}
}}
{{eqn | r = \int \frac {\paren {\cosh x}'} {\cosh x} \rd x
| c = [[Derivative of Hyperbolic Cosine]]
}}
{{eqn | r = \ln \size {\cosh x} + C
| c = [[Primitive of Function ... | Primitive of Hyperbolic Tangent Function/Proof 1 | https://proofwiki.org/wiki/Primitive_of_Hyperbolic_Tangent_Function | https://proofwiki.org/wiki/Primitive_of_Hyperbolic_Tangent_Function/Proof_1 | [
"Primitives of Hyperbolic Functions",
"Hyperbolic Tangent Function",
"Primitive of Hyperbolic Tangent Function"
] | [] | [
"Derivative of Hyperbolic Cosine",
"Primitive of Function under its Derivative",
"Graph of Hyperbolic Cosine Function"
] |
proofwiki-9044 | Primitive of Hyperbolic Tangent Function | :$\ds \int \tanh x \rd x = \map \ln {\cosh x} + C$ | {{begin-eqn}}
{{eqn | l = \int \tanh x \rd x
| r = -i \int \tan i x \rd x
| c = Hyperbolic Tangent in terms of Tangent
}}
{{eqn | r = -\int \tan i x \rd \paren {i x}
| c = Primitive of Function of Constant Multiple
}}
{{eqn | r = \ln \cmod {\cos i x} + C
| c = Primitive of $\tan x$: Cosine Form
}}
{{eqn | r = \ln \... | :$\ds \int \tanh x \rd x = \map \ln {\cosh x} + C$ | {{begin-eqn}}
{{eqn | l = \int \tanh x \rd x
| r = -i \int \tan i x \rd x
| c = [[Hyperbolic Tangent in terms of Tangent]]
}}
{{eqn | r = -\int \tan i x \rd \paren {i x}
| c = [[Primitive of Function of Constant Multiple]]
}}
{{eqn | r = \ln \cmod {\cos i x} + C
| c = [[Primitive of Tangent Function/Cosine Form|Pri... | Primitive of Hyperbolic Tangent Function/Proof 2 | https://proofwiki.org/wiki/Primitive_of_Hyperbolic_Tangent_Function | https://proofwiki.org/wiki/Primitive_of_Hyperbolic_Tangent_Function/Proof_2 | [
"Primitives of Hyperbolic Functions",
"Hyperbolic Tangent Function",
"Primitive of Hyperbolic Tangent Function"
] | [] | [
"Hyperbolic Tangent in terms of Tangent",
"Primitive of Function of Constant Multiple",
"Primitive of Tangent Function/Cosine Form",
"Graph of Hyperbolic Cosine Function"
] |
proofwiki-9045 | Primitive of Hyperbolic Cotangent Function | :$\ds \int \coth x \rd x = \ln \size {\sinh x} + C$
where $\sinh x \ne 0$. | {{begin-eqn}}
{{eqn | l = \int \coth x \rd x
| r = \int \frac {\cosh x} {\sinh x} \rd x
| c = {{Defof|Hyperbolic Cotangent}}
}}
{{eqn | r = \int \frac {\paren {\sinh x}'} {\sinh x} \rd x
| c = Derivative of Hyperbolic Sine
}}
{{eqn | r = \ln \size {\sinh x} + C
| c = Primitive of Function under ... | :$\ds \int \coth x \rd x = \ln \size {\sinh x} + C$
where $\sinh x \ne 0$. | {{begin-eqn}}
{{eqn | l = \int \coth x \rd x
| r = \int \frac {\cosh x} {\sinh x} \rd x
| c = {{Defof|Hyperbolic Cotangent}}
}}
{{eqn | r = \int \frac {\paren {\sinh x}'} {\sinh x} \rd x
| c = [[Derivative of Hyperbolic Sine]]
}}
{{eqn | r = \ln \size {\sinh x} + C
| c = [[Primitive of Function ... | Primitive of Hyperbolic Cotangent Function | https://proofwiki.org/wiki/Primitive_of_Hyperbolic_Cotangent_Function | https://proofwiki.org/wiki/Primitive_of_Hyperbolic_Cotangent_Function | [
"Primitives of Hyperbolic Functions",
"Hyperbolic Cotangent Function",
"Primitives involving Hyperbolic Cotangent Function"
] | [] | [
"Derivative of Hyperbolic Sine",
"Primitive of Function under its Derivative"
] |
proofwiki-9046 | Limit of Tan X over X at Zero | :$\ds \lim_{x \mathop \to 0} \frac {\tan x} x = 1$ | By L'Hôpital's Rule:
{{begin-eqn}}
{{eqn | l = \lim_{x \mathop \to 0} \frac {\tan x} x
| r = \lim_{x \mathop \to 0} \frac {\sec^2 x} 1
| c = Derivative of Tangent Function
}}
{{eqn | r = 1
| c = Secant of Zero
}}
{{end-eqn}}
{{qed}} | :$\ds \lim_{x \mathop \to 0} \frac {\tan x} x = 1$ | By [[L'Hôpital's Rule]]:
{{begin-eqn}}
{{eqn | l = \lim_{x \mathop \to 0} \frac {\tan x} x
| r = \lim_{x \mathop \to 0} \frac {\sec^2 x} 1
| c = [[Derivative of Tangent Function]]
}}
{{eqn | r = 1
| c = [[Secant of Zero]]
}}
{{end-eqn}}
{{qed}} | Limit of Tan X over X at Zero/Proof 1 | https://proofwiki.org/wiki/Limit_of_Tan_X_over_X_at_Zero | https://proofwiki.org/wiki/Limit_of_Tan_X_over_X_at_Zero/Proof_1 | [
"Tangent Function",
"Examples of Limits of Real Functions",
"Limit of Tan X over X at Zero"
] | [] | [
"L'Hôpital's Rule",
"Derivative of Tangent Function",
"Secant of Zero"
] |
proofwiki-9047 | Limit of Tan X over X at Zero | :$\ds \lim_{x \mathop \to 0} \frac {\tan x} x = 1$ | {{begin-eqn}}
{{eqn | l = \lim_{x \mathop \to 0} \frac {\tan x} x
| r = \lim_{x \mathop \to 0} \frac 1 {\cos x} \frac {\sin x} x
| c = {{Defof|Tangent Function}}
}}
{{eqn | r = \lim_{x \mathop \to 0} \frac 1 {\cos x} \lim_{x \mathop \to 0} \frac {\sin x} x
| c = Product Rule for Limits of Real Functi... | :$\ds \lim_{x \mathop \to 0} \frac {\tan x} x = 1$ | {{begin-eqn}}
{{eqn | l = \lim_{x \mathop \to 0} \frac {\tan x} x
| r = \lim_{x \mathop \to 0} \frac 1 {\cos x} \frac {\sin x} x
| c = {{Defof|Tangent Function}}
}}
{{eqn | r = \lim_{x \mathop \to 0} \frac 1 {\cos x} \lim_{x \mathop \to 0} \frac {\sin x} x
| c = [[Product Rule for Limits of Real Func... | Limit of Tan X over X at Zero/Proof 2 | https://proofwiki.org/wiki/Limit_of_Tan_X_over_X_at_Zero | https://proofwiki.org/wiki/Limit_of_Tan_X_over_X_at_Zero/Proof_2 | [
"Tangent Function",
"Examples of Limits of Real Functions",
"Limit of Tan X over X at Zero"
] | [] | [
"Combination Theorem for Limits of Functions/Real/Product Rule",
"Cosine of Zero is One",
"Limit of Sinc Function at Zero"
] |
proofwiki-9048 | Limit of Tan X over X at Zero | :$\ds \lim_{x \mathop \to 0} \frac {\tan x} x = 1$ | Let $f$ be the real function defined as:
:$\map f x = \sin x$
Let:
:$c = \pi$
:$h \in \openint 1 {\dfrac \pi 2}$
We have:
{{begin-eqn}}
{{eqn | l = \map f {c + h}
| r = \map \sin {\pi + h}
| c =
}}
{{eqn | l = \map f c
| r = \sin \pi
| c =
}}
{{eqn | ll= \leadsto
| q = \exists \theta \in... | :$\ds \lim_{x \mathop \to 0} \frac {\tan x} x = 1$ | Let $f$ be the [[Definition:Real Function|real function]] defined as:
:$\map f x = \sin x$
Let:
:$c = \pi$
:$h \in \openint 1 {\dfrac \pi 2}$
We have:
{{begin-eqn}}
{{eqn | l = \map f {c + h}
| r = \map \sin {\pi + h}
| c =
}}
{{eqn | l = \map f c
| r = \sin \pi
| c =
}}
{{eqn | ll= \leadsto... | Limit of Tan X over X at Zero/Proof 3 | https://proofwiki.org/wiki/Limit_of_Tan_X_over_X_at_Zero | https://proofwiki.org/wiki/Limit_of_Tan_X_over_X_at_Zero/Proof_3 | [
"Tangent Function",
"Examples of Limits of Real Functions",
"Limit of Tan X over X at Zero"
] | [] | [
"Definition:Real Function",
"Mean Value Theorem",
"Sine of Angle plus Straight Angle",
"Cosine of Angle plus Straight Angle",
"Squeeze Theorem/Functions"
] |
proofwiki-9049 | Euler Formula for Sine Function/Real Numbers | {{begin-eqn}}
{{eqn | l = \sin x
| r = x \prod_{n \mathop = 1}^\infty \paren {1 - \frac {x^2} {n^2 \pi^2} }
| c =
}}
{{eqn | r = x \paren {1 - \dfrac {x^2} {\pi^2} } \paren {1 - \dfrac {x^2} {4 \pi^2} } \paren {1 - \dfrac {x^2} {9 \pi^2} } \dotsm
| c =
}}
{{end-eqn}}
for all $x \in \R$. | For $x \in \R$ and $n \in \N$, let:
:$\ds \map {I_n} x = \int_0^{\pi / 2} \cos {x t} \cos^n t \rd t $
Observe that:
:$\map {I_0} 0 = \dfrac {\pi} 2$
and:
{{begin-eqn}}
{{eqn | l = \map {I_0} x
| r = \int_0^{\pi / 2} \cos {x t} \rd t
| c =
}}
{{eqn | r = \frac 1 x \map \sin {\frac {\pi x} 2}
| c =
}}... | {{begin-eqn}}
{{eqn | l = \sin x
| r = x \prod_{n \mathop = 1}^\infty \paren {1 - \frac {x^2} {n^2 \pi^2} }
| c =
}}
{{eqn | r = x \paren {1 - \dfrac {x^2} {\pi^2} } \paren {1 - \dfrac {x^2} {4 \pi^2} } \paren {1 - \dfrac {x^2} {9 \pi^2} } \dotsm
| c =
}}
{{end-eqn}}
for all $x \in \R$. | For $x \in \R$ and $n \in \N$, let:
:$\ds \map {I_n} x = \int_0^{\pi / 2} \cos {x t} \cos^n t \rd t $
Observe that:
:$\map {I_0} 0 = \dfrac {\pi} 2$
and:
{{begin-eqn}}
{{eqn | l = \map {I_0} x
| r = \int_0^{\pi / 2} \cos {x t} \rd t
| c =
}}
{{eqn | r = \frac 1 x \map \sin {\frac {\pi x} 2}
| c =... | Euler Formula for Sine Function/Real Numbers/Proof 1 | https://proofwiki.org/wiki/Euler_Formula_for_Sine_Function/Real_Numbers | https://proofwiki.org/wiki/Euler_Formula_for_Sine_Function/Real_Numbers/Proof_1 | [
"Euler Formula for Sine Function"
] | [] | [
"Integration by Parts",
"Sum of Squares of Sine and Cosine",
"Definition:Reduction Formula (Calculus)",
"Shape of Cosine Function",
"Relative Sizes of Definite Integrals",
"Cosine Inequality",
"Tangent Inequality",
"Integration by Parts",
"Squeeze Theorem",
"Definition:Even Integer",
"Principle ... |
proofwiki-9050 | Euler Formula for Sine Function/Real Numbers | {{begin-eqn}}
{{eqn | l = \sin x
| r = x \prod_{n \mathop = 1}^\infty \paren {1 - \frac {x^2} {n^2 \pi^2} }
| c =
}}
{{eqn | r = x \paren {1 - \dfrac {x^2} {\pi^2} } \paren {1 - \dfrac {x^2} {4 \pi^2} } \paren {1 - \dfrac {x^2} {9 \pi^2} } \dotsm
| c =
}}
{{end-eqn}}
for all $x \in \R$. | Using De Moivre's Formula:
:$\sin x = \dfrac {\left({\cos \dfrac x n + i \sin \dfrac x n}\right)^n - \left({\cos \dfrac x n - i \sin \dfrac x n}\right)^n} {2i}$
The difference between two $n$th powers can be extracted into linear factors using $n$th roots of unity.
For large $n$, we can replace:
: $\cos \dfrac x n$ by ... | {{begin-eqn}}
{{eqn | l = \sin x
| r = x \prod_{n \mathop = 1}^\infty \paren {1 - \frac {x^2} {n^2 \pi^2} }
| c =
}}
{{eqn | r = x \paren {1 - \dfrac {x^2} {\pi^2} } \paren {1 - \dfrac {x^2} {4 \pi^2} } \paren {1 - \dfrac {x^2} {9 \pi^2} } \dotsm
| c =
}}
{{end-eqn}}
for all $x \in \R$. | Using [[De Moivre's Formula]]:
:$\sin x = \dfrac {\left({\cos \dfrac x n + i \sin \dfrac x n}\right)^n - \left({\cos \dfrac x n - i \sin \dfrac x n}\right)^n} {2i}$
The difference between two [[Definition:Power (Algebra)|$n$th powers]] can be extracted into linear factors using [[Definition:Complex Roots of Unity|$n$... | Euler Formula for Sine Function/Real Numbers/Proof 2 | https://proofwiki.org/wiki/Euler_Formula_for_Sine_Function/Real_Numbers | https://proofwiki.org/wiki/Euler_Formula_for_Sine_Function/Real_Numbers/Proof_2 | [
"Euler Formula for Sine Function"
] | [] | [
"De Moivre's Formula",
"Definition:Power (Algebra)",
"Definition:Root of Unity/Complex"
] |
proofwiki-9051 | Euler Formula for Sine Function/Real Numbers | {{begin-eqn}}
{{eqn | l = \sin x
| r = x \prod_{n \mathop = 1}^\infty \paren {1 - \frac {x^2} {n^2 \pi^2} }
| c =
}}
{{eqn | r = x \paren {1 - \dfrac {x^2} {\pi^2} } \paren {1 - \dfrac {x^2} {4 \pi^2} } \paren {1 - \dfrac {x^2} {9 \pi^2} } \dotsm
| c =
}}
{{end-eqn}}
for all $x \in \R$. | We have that $\sin x$ has a power series representation:
:$\sin x = x - \dfrac {x^3} {3!} + \dfrac {x^5} {5!} - \dfrac {x^7} {7!} + \cdots$
The roots of sine are the numbers $k \pi$, where $k$ is any integer.
From the Polynomial Factor Theorem, the following ''might'' be true:
:$\ds \sin x = A x \prod \paren {1 - \fra... | {{begin-eqn}}
{{eqn | l = \sin x
| r = x \prod_{n \mathop = 1}^\infty \paren {1 - \frac {x^2} {n^2 \pi^2} }
| c =
}}
{{eqn | r = x \paren {1 - \dfrac {x^2} {\pi^2} } \paren {1 - \dfrac {x^2} {4 \pi^2} } \paren {1 - \dfrac {x^2} {9 \pi^2} } \dotsm
| c =
}}
{{end-eqn}}
for all $x \in \R$. | We have that $\sin x$ has a [[Power Series Expansion for Sine Function|power series representation]]:
:$\sin x = x - \dfrac {x^3} {3!} + \dfrac {x^5} {5!} - \dfrac {x^7} {7!} + \cdots$
The [[Zeroes of Sine and Cosine|roots of sine]] are the numbers $k \pi$, where $k$ is any [[Definition:Integer|integer]].
From the ... | Euler Formula for Sine Function/Real Numbers/Proof 3 | https://proofwiki.org/wiki/Euler_Formula_for_Sine_Function/Real_Numbers | https://proofwiki.org/wiki/Euler_Formula_for_Sine_Function/Real_Numbers/Proof_3 | [
"Euler Formula for Sine Function"
] | [] | [
"Power Series Expansion for Sine Function",
"Zeroes of Sine and Cosine",
"Definition:Integer",
"Polynomial Factor Theorem/Corollary",
"Limit of Sinc Function at Zero"
] |
proofwiki-9052 | Euler Formula for Sine Function/Real Numbers | {{begin-eqn}}
{{eqn | l = \sin x
| r = x \prod_{n \mathop = 1}^\infty \paren {1 - \frac {x^2} {n^2 \pi^2} }
| c =
}}
{{eqn | r = x \paren {1 - \dfrac {x^2} {\pi^2} } \paren {1 - \dfrac {x^2} {4 \pi^2} } \paren {1 - \dfrac {x^2} {9 \pi^2} } \dotsm
| c =
}}
{{end-eqn}}
for all $x \in \R$. | For $x \in \R$ and $n \in \N_{> 0}$, let:
:$\map {f_n} x = \dfrac 1 2 \paren {\paren {1 + \dfrac x n}^n - \paren {1 - \dfrac x n}^n }$
Then $\map {f_n} x = 0$ {{iff}}:
{{begin-eqn}}
{{eqn | l = \paren {1 + \frac x n}^n
| r = \paren {1 - \frac x n}^n
| c =
}}
{{eqn | ll= \leadstoandfrom
| l = 1 + \fra... | {{begin-eqn}}
{{eqn | l = \sin x
| r = x \prod_{n \mathop = 1}^\infty \paren {1 - \frac {x^2} {n^2 \pi^2} }
| c =
}}
{{eqn | r = x \paren {1 - \dfrac {x^2} {\pi^2} } \paren {1 - \dfrac {x^2} {4 \pi^2} } \paren {1 - \dfrac {x^2} {9 \pi^2} } \dotsm
| c =
}}
{{end-eqn}}
for all $x \in \R$. | For $x \in \R$ and $n \in \N_{> 0}$, let:
:$\map {f_n} x = \dfrac 1 2 \paren {\paren {1 + \dfrac x n}^n - \paren {1 - \dfrac x n}^n }$
Then $\map {f_n} x = 0$ {{iff}}:
{{begin-eqn}}
{{eqn | l = \paren {1 + \frac x n}^n
| r = \paren {1 - \frac x n}^n
| c =
}}
{{eqn | ll= \leadstoandfrom
| l = 1 + \... | Euler Formula for Sine Function/Real Numbers/Proof 4 | https://proofwiki.org/wiki/Euler_Formula_for_Sine_Function/Real_Numbers | https://proofwiki.org/wiki/Euler_Formula_for_Sine_Function/Real_Numbers/Proof_4 | [
"Euler Formula for Sine Function"
] | [] | [
"Euler's Tangent Identity",
"Definition:Root of Polynomial",
"Definition:Polynomial/Real Numbers",
"Definition:Degree of Polynomial",
"Definition:Constant",
"Polynomial Factor Theorem",
"Tangent Function is Odd",
"Binomial Theorem/Integral Index",
"Limit of Tan X over X at Zero",
"Tangent Inequali... |
proofwiki-9053 | Euler Formula for Sine Function/Complex Numbers | {{begin-eqn}}
{{eqn | l = \sin z
| r = z \prod_{n \mathop = 1}^\infty \paren {1 - \frac {z^2} {n^2 \pi^2} }
| c =
}}
{{eqn | r = z \paren {1 - \dfrac {z^2} {\pi^2} } \paren {1 - \dfrac {z^2} {4 \pi^2} } \paren {1 - \dfrac {z^2} {9 \pi^2} } \dotsm
| c =
}}
{{end-eqn}}
for all $z \in \C$. | For $z \in \C$ and $n \in \N$, let:
:$\ds \map {I_n} z = \int_0^{\pi / 2} \cos {z t} \cos^n t \rd t $
Observe that $\map {I_0} 0 = \dfrac {\pi} 2$ and:
{{begin-eqn}}
{{eqn | l = \map {I_0} z
| r = \int_0^{\pi / 2} \cos {z t} \rd t
| c =
}}
{{eqn | r = \frac 1 z \map \sin {\frac {\pi z} 2}
| c =
}}
{... | {{begin-eqn}}
{{eqn | l = \sin z
| r = z \prod_{n \mathop = 1}^\infty \paren {1 - \frac {z^2} {n^2 \pi^2} }
| c =
}}
{{eqn | r = z \paren {1 - \dfrac {z^2} {\pi^2} } \paren {1 - \dfrac {z^2} {4 \pi^2} } \paren {1 - \dfrac {z^2} {9 \pi^2} } \dotsm
| c =
}}
{{end-eqn}}
for all $z \in \C$. | For $z \in \C$ and $n \in \N$, let:
:$\ds \map {I_n} z = \int_0^{\pi / 2} \cos {z t} \cos^n t \rd t $
Observe that $\map {I_0} 0 = \dfrac {\pi} 2$ and:
{{begin-eqn}}
{{eqn | l = \map {I_0} z
| r = \int_0^{\pi / 2} \cos {z t} \rd t
| c =
}}
{{eqn | r = \frac 1 z \map \sin {\frac {\pi z} 2}
| c =
}... | Euler Formula for Sine Function/Complex Numbers/Proof 1 | https://proofwiki.org/wiki/Euler_Formula_for_Sine_Function/Complex_Numbers | https://proofwiki.org/wiki/Euler_Formula_for_Sine_Function/Complex_Numbers/Proof_1 | [
"Euler Formula for Sine Function"
] | [] | [
"Integration by Parts",
"Sum of Squares of Sine and Cosine",
"Definition:Reduction Formula (Calculus)",
"Shape of Cosine Function",
"Cosine of Sum",
"Hyperbolic Sine in terms of Sine",
"Hyperbolic Cosine in terms of Cosine",
"Double Angle Formulas",
"Sine Inequality",
"Euler Formula for Sine Funct... |
proofwiki-9054 | Euler Formula for Sine Function/Complex Numbers | {{begin-eqn}}
{{eqn | l = \sin z
| r = z \prod_{n \mathop = 1}^\infty \paren {1 - \frac {z^2} {n^2 \pi^2} }
| c =
}}
{{eqn | r = z \paren {1 - \dfrac {z^2} {\pi^2} } \paren {1 - \dfrac {z^2} {4 \pi^2} } \paren {1 - \dfrac {z^2} {9 \pi^2} } \dotsm
| c =
}}
{{end-eqn}}
for all $z \in \C$. | For $z \in \C$ and $n \in \N_{> 0}$, let:
:$\map {f_n} z = \dfrac 1 2 \paren {\paren {1 + \dfrac z n}^n - \paren {1 - \dfrac z n}^n}$
Then $\map {f_n} z = 0$ {{iff}}:
{{begin-eqn}}
{{eqn | l = \paren {1 + \dfrac z n}^n
| r = \paren {1 - \dfrac z n}^n
| c =
}}
{{eqn | ll= \leadstoandfrom
| l = 1 + \fr... | {{begin-eqn}}
{{eqn | l = \sin z
| r = z \prod_{n \mathop = 1}^\infty \paren {1 - \frac {z^2} {n^2 \pi^2} }
| c =
}}
{{eqn | r = z \paren {1 - \dfrac {z^2} {\pi^2} } \paren {1 - \dfrac {z^2} {4 \pi^2} } \paren {1 - \dfrac {z^2} {9 \pi^2} } \dotsm
| c =
}}
{{end-eqn}}
for all $z \in \C$. | For $z \in \C$ and $n \in \N_{> 0}$, let:
:$\map {f_n} z = \dfrac 1 2 \paren {\paren {1 + \dfrac z n}^n - \paren {1 - \dfrac z n}^n}$
Then $\map {f_n} z = 0$ {{iff}}:
{{begin-eqn}}
{{eqn | l = \paren {1 + \dfrac z n}^n
| r = \paren {1 - \dfrac z n}^n
| c =
}}
{{eqn | ll= \leadstoandfrom
| l = 1 + ... | Euler Formula for Sine Function/Complex Numbers/Proof 2 | https://proofwiki.org/wiki/Euler_Formula_for_Sine_Function/Complex_Numbers | https://proofwiki.org/wiki/Euler_Formula_for_Sine_Function/Complex_Numbers/Proof_2 | [
"Euler Formula for Sine Function"
] | [] | [
"Euler's Tangent Identity",
"Definition:Root of Polynomial",
"Definition:Degree of Polynomial",
"Polynomial Factor Theorem",
"Tangent Function is Odd",
"Binomial Theorem/Integral Index",
"Definition:Positive/Real Number",
"Limit of Tan X over X at Zero",
"Tangent Inequality",
"Squeeze Theorem",
... |
proofwiki-9055 | Logarithm of Reciprocal | :$\map {\log_b} {\dfrac 1 x} = -\log_b x$ | {{begin-eqn}}
{{eqn | l = \map {\log_b} {\dfrac 1 x}
| r = \log_b 1 - \log_b x
| c = Difference of Logarithms
}}
{{eqn | r = 0 - \log_b x
| c = Logarithm of 1 is 0
}}
{{end-eqn}}
Hence the result.
{{qed}}
Category:Logarithms
Category:Reciprocals
kgfbjw7h1n6cupqiu7qtvossewt0xp8 | :$\map {\log_b} {\dfrac 1 x} = -\log_b x$ | {{begin-eqn}}
{{eqn | l = \map {\log_b} {\dfrac 1 x}
| r = \log_b 1 - \log_b x
| c = [[Difference of Logarithms]]
}}
{{eqn | r = 0 - \log_b x
| c = [[Logarithm of 1 is 0]]
}}
{{end-eqn}}
Hence the result.
{{qed}}
[[Category:Logarithms]]
[[Category:Reciprocals]]
kgfbjw7h1n6cupqiu7qtvossewt0xp8 | Logarithm of Reciprocal | https://proofwiki.org/wiki/Logarithm_of_Reciprocal | https://proofwiki.org/wiki/Logarithm_of_Reciprocal | [
"Logarithms",
"Reciprocals"
] | [] | [
"Difference of Logarithms",
"Natural Logarithm of 1 is 0",
"Category:Logarithms",
"Category:Reciprocals"
] |
proofwiki-9056 | Primitive of Hyperbolic Secant Function/Arcsine Form | :$\ds \int \sech x \rd x = \map \arcsin {\tanh x} + C$ | Let:
{{begin-eqn}}
{{eqn | l = \int \sech x \rd x
| r = \int \frac {\sech^2 x} {\sech x} \rd x
| c =
}}
{{eqn | r = \int \frac {\sech^2 x} {\sqrt {1 - \tanh^2 x} } \rd x
| c = Sum of Squares of Hyperbolic Secant and Tangent
}}
{{end-eqn}}
Let:
{{begin-eqn}}
{{eqn | l = u
| r = \tanh x
| c... | :$\ds \int \sech x \rd x = \map \arcsin {\tanh x} + C$ | Let:
{{begin-eqn}}
{{eqn | l = \int \sech x \rd x
| r = \int \frac {\sech^2 x} {\sech x} \rd x
| c =
}}
{{eqn | r = \int \frac {\sech^2 x} {\sqrt {1 - \tanh^2 x} } \rd x
| c = [[Sum of Squares of Hyperbolic Secant and Tangent]]
}}
{{end-eqn}}
Let:
{{begin-eqn}}
{{eqn | l = u
| r = \tanh x
... | Primitive of Hyperbolic Secant Function/Arcsine Form | https://proofwiki.org/wiki/Primitive_of_Hyperbolic_Secant_Function/Arcsine_Form | https://proofwiki.org/wiki/Primitive_of_Hyperbolic_Secant_Function/Arcsine_Form | [
"Primitive of Hyperbolic Secant Function"
] | [] | [
"Sum of Squares of Hyperbolic Secant and Tangent",
"Derivative of Hyperbolic Tangent",
"Integration by Substitution",
"Primitive of Reciprocal of Root of a squared minus x squared/Arcsine Form"
] |
proofwiki-9057 | Primitive of Hyperbolic Secant Function/Arctangent of Exponential Form | :$\ds \int \sech x \rd x = 2 \map \arctan {e^x} + C$ | Let:
{{begin-eqn}}
{{eqn | l = \int \sech x \rd x
| r = \int \frac 2 {e^x + e^{-x} } \rd x
| c = {{Defof|Hyperbolic Secant|index = 1}}
}}
{{eqn | r = \int \frac {2 e^x} {e^{2 x} + 1} \rd x
| c = multiplying top and bottom by $e^x$
}}
{{end-eqn}}
Let:
{{begin-eqn}}
{{eqn | l = u
| r = e^x
|... | :$\ds \int \sech x \rd x = 2 \map \arctan {e^x} + C$ | Let:
{{begin-eqn}}
{{eqn | l = \int \sech x \rd x
| r = \int \frac 2 {e^x + e^{-x} } \rd x
| c = {{Defof|Hyperbolic Secant|index = 1}}
}}
{{eqn | r = \int \frac {2 e^x} {e^{2 x} + 1} \rd x
| c = multiplying [[Definition:Numerator|top]] and [[Definition:Denominator|bottom]] by $e^x$
}}
{{end-eqn}}
Let... | Primitive of Hyperbolic Secant Function/Arctangent of Exponential Form | https://proofwiki.org/wiki/Primitive_of_Hyperbolic_Secant_Function/Arctangent_of_Exponential_Form | https://proofwiki.org/wiki/Primitive_of_Hyperbolic_Secant_Function/Arctangent_of_Exponential_Form | [
"Primitive of Hyperbolic Secant Function"
] | [] | [
"Definition:Fraction/Numerator",
"Definition:Fraction/Denominator",
"Derivative of Exponential Function",
"Integration by Substitution",
"Primitive of Reciprocal of x squared plus a squared/Arctangent Form"
] |
proofwiki-9058 | Primitive of Hyperbolic Cosecant Function/Logarithm Form | :$\ds \int \csch x \rd x = -\ln \size {\csch x + \coth x} + C$
where $\csch x + \coth x \ne 0$. | Let:
{{begin-eqn}}
{{eqn | l = u
| r = \coth x + \csch x
| c =
}}
{{eqn | ll= \leadsto
| l = \frac {\d u} {\d x}
| r = \frac \d {\d x} \coth x + \frac \d {\d x} \csch x
| c = Linear Combination of Derivatives
}}
{{eqn | r = -\csch^2 x + \frac \d {\d x} \csch x
| c = Derivative of ... | :$\ds \int \csch x \rd x = -\ln \size {\csch x + \coth x} + C$
where $\csch x + \coth x \ne 0$. | Let:
{{begin-eqn}}
{{eqn | l = u
| r = \coth x + \csch x
| c =
}}
{{eqn | ll= \leadsto
| l = \frac {\d u} {\d x}
| r = \frac \d {\d x} \coth x + \frac \d {\d x} \csch x
| c = [[Linear Combination of Derivatives]]
}}
{{eqn | r = -\csch^2 x + \frac \d {\d x} \csch x
| c = [[Derivati... | Primitive of Hyperbolic Cosecant Function/Logarithm Form | https://proofwiki.org/wiki/Primitive_of_Hyperbolic_Cosecant_Function/Logarithm_Form | https://proofwiki.org/wiki/Primitive_of_Hyperbolic_Cosecant_Function/Logarithm_Form | [
"Primitive of Hyperbolic Cosecant Function"
] | [] | [
"Linear Combination of Derivatives",
"Derivative of Hyperbolic Cotangent",
"Derivative of Hyperbolic Cosecant",
"Definition:Fraction/Numerator",
"Definition:Fraction/Denominator",
"Primitive of Constant Multiple of Function",
"Primitive of Function under its Derivative"
] |
proofwiki-9059 | Primitive of Hyperbolic Cosecant Function/Hyperbolic Tangent Form | :$\ds \int \csch x \rd x = \ln \size {\tanh \frac x 2} + C$
where $\tanh \dfrac x 2 \ne 0$. | {{begin-eqn}}
{{eqn | l = \int \csch x \rd x
| r = -\ln \size {\csch x + \coth x} + C
| c = Primitive of $\csch x$: Logarithm Form
}}
{{eqn | r = \ln \size {\frac 1 {\csch x + \coth x} } + C
| c = Logarithm of Reciprocal
}}
{{eqn | r = \ln \size {\frac 1 {\frac 1 {\sinh x} + \frac {\cosh x} {\sinh x} ... | :$\ds \int \csch x \rd x = \ln \size {\tanh \frac x 2} + C$
where $\tanh \dfrac x 2 \ne 0$. | {{begin-eqn}}
{{eqn | l = \int \csch x \rd x
| r = -\ln \size {\csch x + \coth x} + C
| c = [[Primitive of Hyperbolic Cosecant Function/Logarithm Form|Primitive of $\csch x$: Logarithm Form]]
}}
{{eqn | r = \ln \size {\frac 1 {\csch x + \coth x} } + C
| c = [[Logarithm of Reciprocal]]
}}
{{eqn | r = \... | Primitive of Hyperbolic Cosecant Function/Hyperbolic Tangent Form/Proof 2 | https://proofwiki.org/wiki/Primitive_of_Hyperbolic_Cosecant_Function/Hyperbolic_Tangent_Form | https://proofwiki.org/wiki/Primitive_of_Hyperbolic_Cosecant_Function/Hyperbolic_Tangent_Form/Proof_2 | [
"Primitive of Hyperbolic Cosecant Function"
] | [] | [
"Primitive of Hyperbolic Cosecant Function/Logarithm Form",
"Logarithm of Reciprocal"
] |
proofwiki-9060 | Primitive of Hyperbolic Cosecant Function/Inverse Hyperbolic Cotangent Form | :$\ds \int \csch x \rd x = -2 \map {\coth^{-1} } {e^x} + C$ | Let:
{{begin-eqn}}
{{eqn | l = \int \csch x \rd x
| r = \int \frac 2 {e^x - e^{-x} } \rd x
| c = {{Defof|Hyperbolic Cosecant}}
}}
{{eqn | r = \int \frac {2 e^x} {e^{2 x} - 1} \rd x
| c = multiplying top and bottom by $e^x$
}}
{{end-eqn}}
Let:
{{begin-eqn}}
{{eqn | l = u
| r = e^x
| c =
}}... | :$\ds \int \csch x \rd x = -2 \map {\coth^{-1} } {e^x} + C$ | Let:
{{begin-eqn}}
{{eqn | l = \int \csch x \rd x
| r = \int \frac 2 {e^x - e^{-x} } \rd x
| c = {{Defof|Hyperbolic Cosecant}}
}}
{{eqn | r = \int \frac {2 e^x} {e^{2 x} - 1} \rd x
| c = multiplying [[Definition:Numerator|top]] and [[Definition:Denominator|bottom]] by $e^x$
}}
{{end-eqn}}
Let:
{{begi... | Primitive of Hyperbolic Cosecant Function/Inverse Hyperbolic Cotangent Form | https://proofwiki.org/wiki/Primitive_of_Hyperbolic_Cosecant_Function/Inverse_Hyperbolic_Cotangent_Form | https://proofwiki.org/wiki/Primitive_of_Hyperbolic_Cosecant_Function/Inverse_Hyperbolic_Cotangent_Form | [
"Primitive of Hyperbolic Cosecant Function",
"Primitives involving Hyperbolic Cotangent Function"
] | [] | [
"Definition:Fraction/Numerator",
"Definition:Fraction/Denominator",
"Derivative of Exponential Function",
"Integration by Substitution",
"Primitive of Reciprocal of x squared minus a squared/Inverse Hyperbolic Cotangent Form"
] |
proofwiki-9061 | Primitive of Square of Hyperbolic Secant Function | :$\ds \int \sech^2 x \rd x = \tanh x + C$
where $C$ is an arbitrary constant. | From Derivative of Hyperbolic Tangent:
:$\map {\dfrac \d {\d x} } {\tanh x} = \sech^2 x$
The result follows from the definition of primitive.
{{Qed}} | :$\ds \int \sech^2 x \rd x = \tanh x + C$
where $C$ is an [[Definition:Arbitrary Constant (Calculus)|arbitrary constant]]. | From [[Derivative of Hyperbolic Tangent]]:
:$\map {\dfrac \d {\d x} } {\tanh x} = \sech^2 x$
The result follows from the definition of [[Definition:Primitive (Calculus)|primitive]].
{{Qed}} | Primitive of Square of Hyperbolic Secant Function | https://proofwiki.org/wiki/Primitive_of_Square_of_Hyperbolic_Secant_Function | https://proofwiki.org/wiki/Primitive_of_Square_of_Hyperbolic_Secant_Function | [
"Primitives of Hyperbolic Functions",
"Hyperbolic Secant Function",
"Hyperbolic Tangent Function"
] | [
"Definition:Primitive (Calculus)/Constant of Integration"
] | [
"Derivative of Hyperbolic Tangent",
"Definition:Primitive (Calculus)"
] |
proofwiki-9062 | Primitive of Square of Hyperbolic Cosecant Function | :$\ds \int \csch^2 x \rd x = -\coth x + C$
where $C$ is an arbitrary constant. | From Derivative of Hyperbolic Cotangent:
:$\map {\dfrac \d {\d x} } {\coth x} = -\csch^2 x$
The result follows from the definition of primitive.
{{Qed}} | :$\ds \int \csch^2 x \rd x = -\coth x + C$
where $C$ is an [[Definition:Arbitrary Constant (Calculus)|arbitrary constant]]. | From [[Derivative of Hyperbolic Cotangent]]:
:$\map {\dfrac \d {\d x} } {\coth x} = -\csch^2 x$
The result follows from the definition of [[Definition:Primitive (Calculus)|primitive]].
{{Qed}} | Primitive of Square of Hyperbolic Cosecant Function | https://proofwiki.org/wiki/Primitive_of_Square_of_Hyperbolic_Cosecant_Function | https://proofwiki.org/wiki/Primitive_of_Square_of_Hyperbolic_Cosecant_Function | [
"Primitives of Hyperbolic Functions",
"Hyperbolic Cosecant Function",
"Hyperbolic Cotangent Function"
] | [
"Definition:Primitive (Calculus)/Constant of Integration"
] | [
"Derivative of Hyperbolic Cotangent",
"Definition:Primitive (Calculus)"
] |
proofwiki-9063 | Primitive of Square of Hyperbolic Tangent Function | :$\ds \int \tanh^2 x \rd x = x - \tanh x + C$
where $C$ is an arbitrary constant. | {{begin-eqn}}
{{eqn | l = \int \tanh^2 x \rd x
| r = \int \paren {1 - \sech^2 x} \rd x
| c = Sum of Squares of Hyperbolic Secant and Tangent
}}
{{eqn | r = \int 1 \rd x - \int \sech^2 x \rd x
| c = Linear Combination of Primitives
}}
{{eqn | r = \int 1 \rd x - \tanh x + C
| c = Primitive of Squa... | :$\ds \int \tanh^2 x \rd x = x - \tanh x + C$
where $C$ is an [[Definition:Arbitrary Constant (Calculus)|arbitrary constant]]. | {{begin-eqn}}
{{eqn | l = \int \tanh^2 x \rd x
| r = \int \paren {1 - \sech^2 x} \rd x
| c = [[Sum of Squares of Hyperbolic Secant and Tangent]]
}}
{{eqn | r = \int 1 \rd x - \int \sech^2 x \rd x
| c = [[Linear Combination of Primitives]]
}}
{{eqn | r = \int 1 \rd x - \tanh x + C
| c = [[Primiti... | Primitive of Square of Hyperbolic Tangent Function | https://proofwiki.org/wiki/Primitive_of_Square_of_Hyperbolic_Tangent_Function | https://proofwiki.org/wiki/Primitive_of_Square_of_Hyperbolic_Tangent_Function | [
"Primitives of Hyperbolic Functions",
"Hyperbolic Tangent Function"
] | [
"Definition:Primitive (Calculus)/Constant of Integration"
] | [
"Sum of Squares of Hyperbolic Secant and Tangent",
"Linear Combination of Integrals/Indefinite",
"Primitive of Square of Hyperbolic Secant Function",
"Primitive of Constant"
] |
proofwiki-9064 | Primitive of Square of Hyperbolic Cotangent Function | :$\ds \int \coth^2 x \rd x = x - \coth x + C$
where $C$ is an arbitrary constant. | {{begin-eqn}}
{{eqn | l = \int \coth^2 x \rd x
| r = \int \paren {1 + \csch^2 x} \rd x
| c = Difference of Squares of Hyperbolic Cotangent and Cosecant
}}
{{eqn | r = \int 1 \rd x + \int \csch^2 x \rd x
| c = Linear Combination of Primitives
}}
{{eqn | r = \int 1 \rd x - \coth x + C
| c = Primit... | :$\ds \int \coth^2 x \rd x = x - \coth x + C$
where $C$ is an [[Definition:Arbitrary Constant (Calculus)|arbitrary constant]]. | {{begin-eqn}}
{{eqn | l = \int \coth^2 x \rd x
| r = \int \paren {1 + \csch^2 x} \rd x
| c = [[Difference of Squares of Hyperbolic Cotangent and Cosecant]]
}}
{{eqn | r = \int 1 \rd x + \int \csch^2 x \rd x
| c = [[Linear Combination of Primitives]]
}}
{{eqn | r = \int 1 \rd x - \coth x + C
| c ... | Primitive of Square of Hyperbolic Cotangent Function | https://proofwiki.org/wiki/Primitive_of_Square_of_Hyperbolic_Cotangent_Function | https://proofwiki.org/wiki/Primitive_of_Square_of_Hyperbolic_Cotangent_Function | [
"Primitives of Hyperbolic Functions",
"Hyperbolic Cotangent Function"
] | [
"Definition:Primitive (Calculus)/Constant of Integration"
] | [
"Difference of Squares of Hyperbolic Cotangent and Cosecant",
"Linear Combination of Integrals/Indefinite",
"Primitive of Square of Hyperbolic Cosecant Function",
"Primitive of Constant"
] |
proofwiki-9065 | Heaviside Step Function is Piecewise Continuous | Let $c \ge 0$ be a constant real number.
The Heaviside step function:
:$\map {\mu_c} t = \begin {cases} 1 & : t > c \\ 0 & : t < c \end {cases}$
is piecewise continuous for any interval of the form:
:$\closedint {c - M} {c + M}$
where $M > 0$ is some arbitrarily large constant. | Let the finite subdivision of $\mu_c$ be:
:$\set {c - M, c, c + M}$
Let $\epsilon > 0$.
For $t \to \paren {c - M}^+$, choose $\delta$ to be any number at all, because:
:$c - M < t < c - M + \delta \implies \size {\map {\mu_c} t - 0} = \size {0 - 0} < \epsilon$
holds, from True Statement is implied by Every Statement.
T... | Let $c \ge 0$ be a [[Definition:Constant|constant]] [[Definition:Real Number|real number]].
The [[Definition:Heaviside Step Function|Heaviside step function]]:
:$\map {\mu_c} t = \begin {cases} 1 & : t > c \\ 0 & : t < c \end {cases}$
is [[Definition:Piecewise Continuous Function|piecewise continuous]] for any [[De... | Let the [[Definition:Finite Set|finite]] [[Definition:Subdivision of Interval|subdivision]] of $\mu_c$ be:
:$\set {c - M, c, c + M}$
Let $\epsilon > 0$.
For $t \to \paren {c - M}^+$, choose $\delta$ to be any number at all, because:
:$c - M < t < c - M + \delta \implies \size {\map {\mu_c} t - 0} = \size {0 - 0} <... | Heaviside Step Function is Piecewise Continuous | https://proofwiki.org/wiki/Heaviside_Step_Function_is_Piecewise_Continuous | https://proofwiki.org/wiki/Heaviside_Step_Function_is_Piecewise_Continuous | [
"Heaviside Step Function",
"Continuous Functions"
] | [
"Definition:Constant",
"Definition:Real Number",
"Definition:Heaviside Step Function",
"Definition:Piecewise Continuous Function",
"Definition:Real Interval/Closed",
"Definition:Arbitrarily Large",
"Definition:Constant"
] | [
"Definition:Finite Set",
"Definition:Subdivision of Interval",
"True Statement is implied by Every Statement",
"True Statement is implied by Every Statement",
"Continuity of Heaviside Step Function",
"Category:Heaviside Step Function",
"Category:Continuous Functions"
] |
proofwiki-9066 | Euler Formula for Sine Function/Complex Numbers/Proof 1/Lemma 1 | The function:
:$\dfrac {\sinh x} x$
is increasing for positive real $x$. | Let $\map f x = \dfrac {\sinh x} x$.
By Quotient Rule for Derivatives and Derivative of Hyperbolic Sine:
:$\map {f'} x = \dfrac {x \cosh x - \sinh x} {x^2}$
From Hyperbolic Tangent Less than X, we have $\tanh x \le x$ for $x \ge 0$.
Since $\cosh x \ge 0$, we can rearrange to get $x \cosh x - \sinh x \ge 0$.
Since $x^2 ... | The function:
:$\dfrac {\sinh x} x$
is [[Definition:Increasing Real Function|increasing]] for [[Definition:Positive Real Number|positive real]] $x$. | Let $\map f x = \dfrac {\sinh x} x$.
By [[Quotient Rule for Derivatives]] and [[Derivative of Hyperbolic Sine]]:
:$\map {f'} x = \dfrac {x \cosh x - \sinh x} {x^2}$
From [[Hyperbolic Tangent Less than X]], we have $\tanh x \le x$ for $x \ge 0$.
Since $\cosh x \ge 0$, we can rearrange to get $x \cosh x - \sinh x \ge... | Euler Formula for Sine Function/Complex Numbers/Proof 1/Lemma 1 | https://proofwiki.org/wiki/Euler_Formula_for_Sine_Function/Complex_Numbers/Proof_1/Lemma_1 | https://proofwiki.org/wiki/Euler_Formula_for_Sine_Function/Complex_Numbers/Proof_1/Lemma_1 | [
"Hyperbolic Sine Function",
"Euler Formula for Sine Function"
] | [
"Definition:Increasing/Real Function",
"Definition:Positive/Real Number"
] | [
"Quotient Rule for Derivatives",
"Derivative of Hyperbolic Sine",
"Hyperbolic Tangent Less than X",
"Derivative of Monotone Function",
"Definition:Increasing/Real Function",
"Category:Hyperbolic Sine Function",
"Category:Euler Formula for Sine Function"
] |
proofwiki-9067 | Hyperbolic Tangent Less than X | :$\tanh x \le x$
for $x \ge 0$. | Let $\map f x = x - \tanh x$.
By Derivative of Hyperbolic Tangent:
:$\map {f'} x = 1 - \sech^2 x$
Since $\cosh x \ge 1$ for all $x \in \R$, we can deduce that:
:$\map {f'} x \ge 0$
From Derivative of Monotone Function, $\map f x$ is increasing.
By definition of hyperbolic tangent:
:$\map f x = 0$
It follows that $\map ... | :$\tanh x \le x$
for $x \ge 0$. | Let $\map f x = x - \tanh x$.
By [[Derivative of Hyperbolic Tangent]]:
:$\map {f'} x = 1 - \sech^2 x$
Since $\cosh x \ge 1$ for all $x \in \R$, we can deduce that:
:$\map {f'} x \ge 0$
From [[Derivative of Monotone Function]], $\map f x$ is [[Definition:Increasing Real Function|increasing]].
By definition of [[Defi... | Hyperbolic Tangent Less than X | https://proofwiki.org/wiki/Hyperbolic_Tangent_Less_than_X | https://proofwiki.org/wiki/Hyperbolic_Tangent_Less_than_X | [
"Hyperbolic Tangent Function"
] | [] | [
"Derivative of Hyperbolic Tangent",
"Derivative of Monotone Function",
"Definition:Increasing/Real Function",
"Definition:Hyperbolic Tangent",
"Category:Hyperbolic Tangent Function"
] |
proofwiki-9068 | Primitive of Square of Hyperbolic Sine Function | :$\ds \int \sinh^2 x \rd x = \frac {\sinh 2 x} 4 - \frac x 2 + C$ | {{begin-eqn}}
{{eqn | l = \int \sinh^2 x \rd x
| r = \int \paren {\frac {\cosh 2 x - 1} 2} \rd x
| c = Square of Hyperbolic Sine
}}
{{eqn | r = \int \paren {\frac {\cosh 2 x} 2} \rd x - \int \frac 1 2 \rd x
| c = Linear Combination of Primitives
}}
{{eqn | r = \int \paren {\frac {\cosh 2 x} 2} \rd x -... | :$\ds \int \sinh^2 x \rd x = \frac {\sinh 2 x} 4 - \frac x 2 + C$ | {{begin-eqn}}
{{eqn | l = \int \sinh^2 x \rd x
| r = \int \paren {\frac {\cosh 2 x - 1} 2} \rd x
| c = [[Square of Hyperbolic Sine]]
}}
{{eqn | r = \int \paren {\frac {\cosh 2 x} 2} \rd x - \int \frac 1 2 \rd x
| c = [[Linear Combination of Primitives]]
}}
{{eqn | r = \int \paren {\frac {\cosh 2 x} 2}... | Primitive of Square of Hyperbolic Sine Function | https://proofwiki.org/wiki/Primitive_of_Square_of_Hyperbolic_Sine_Function | https://proofwiki.org/wiki/Primitive_of_Square_of_Hyperbolic_Sine_Function | [
"Primitive of Square of Hyperbolic Sine Function",
"Primitives of Hyperbolic Functions",
"Hyperbolic Sine Function"
] | [] | [
"Power Reduction Formulas/Hyperbolic Sine Squared",
"Linear Combination of Integrals/Indefinite",
"Primitive of Constant",
"Primitive of Function of Constant Multiple",
"Primitive of Hyperbolic Cosine Function"
] |
proofwiki-9069 | Primitive of Square of Hyperbolic Cosine Function | :$\ds \int \cosh^2 x \rd x = \frac {\sinh 2 x} 4 + \frac x 2 + C$ | {{begin-eqn}}
{{eqn | l = \int \cosh^2 x \rd x
| r = \int \paren {\frac {\cosh 2 x + 1} 2} \rd x
| c = Square of Hyperbolic Cosine
}}
{{eqn | r = \int \paren {\frac {\cosh 2 x} 2} \rd x + \int \frac 1 2 \rd x
| c = Linear Combination of Primitives
}}
{{eqn | r = \int \paren {\frac {\cosh 2 x} 2} \rd x... | :$\ds \int \cosh^2 x \rd x = \frac {\sinh 2 x} 4 + \frac x 2 + C$ | {{begin-eqn}}
{{eqn | l = \int \cosh^2 x \rd x
| r = \int \paren {\frac {\cosh 2 x + 1} 2} \rd x
| c = [[Square of Hyperbolic Cosine]]
}}
{{eqn | r = \int \paren {\frac {\cosh 2 x} 2} \rd x + \int \frac 1 2 \rd x
| c = [[Linear Combination of Primitives]]
}}
{{eqn | r = \int \paren {\frac {\cosh 2 x} ... | Primitive of Square of Hyperbolic Cosine Function | https://proofwiki.org/wiki/Primitive_of_Square_of_Hyperbolic_Cosine_Function | https://proofwiki.org/wiki/Primitive_of_Square_of_Hyperbolic_Cosine_Function | [
"Primitive of Square of Hyperbolic Cosine Function",
"Primitives of Hyperbolic Functions",
"Hyperbolic Cosine Function"
] | [] | [
"Power Reduction Formulas/Hyperbolic Cosine Squared",
"Linear Combination of Integrals/Indefinite",
"Primitive of Constant",
"Primitive of Function of Constant Multiple",
"Primitive of Hyperbolic Cosine Function"
] |
proofwiki-9070 | Primitive of Product of Hyperbolic Secant and Tangent | :$\ds \int \sech x \tanh x \rd x = -\sech x + C$
where $C$ is an arbitrary constant. | From Derivative of Hyperbolic Secant:
:$\dfrac \d {\d x} \sech x = -\sech x \tanh x$
The result follows from the definition of primitive.
{{Qed}} | :$\ds \int \sech x \tanh x \rd x = -\sech x + C$
where $C$ is an [[Definition:Arbitrary Constant (Calculus)|arbitrary constant]]. | From [[Derivative of Hyperbolic Secant]]:
:$\dfrac \d {\d x} \sech x = -\sech x \tanh x$
The result follows from the definition of [[Definition:Primitive (Calculus)|primitive]].
{{Qed}} | Primitive of Product of Hyperbolic Secant and Tangent | https://proofwiki.org/wiki/Primitive_of_Product_of_Hyperbolic_Secant_and_Tangent | https://proofwiki.org/wiki/Primitive_of_Product_of_Hyperbolic_Secant_and_Tangent | [
"Primitives of Hyperbolic Functions",
"Hyperbolic Secant Function",
"Hyperbolic Tangent Function"
] | [
"Definition:Primitive (Calculus)/Constant of Integration"
] | [
"Derivative of Hyperbolic Secant",
"Definition:Primitive (Calculus)"
] |
proofwiki-9071 | Primitive of Product of Hyperbolic Cosecant and Cotangent | :$\ds \int \csch x \coth x \rd x = -\csch x + C$
where $C$ is an arbitrary constant. | From Derivative of Hyperbolic Cosecant:
:$\dfrac \d {\d x} \csch x = -\csch x \coth x$
The result follows from the definition of primitive.
{{Qed}} | :$\ds \int \csch x \coth x \rd x = -\csch x + C$
where $C$ is an [[Definition:Arbitrary Constant (Calculus)|arbitrary constant]]. | From [[Derivative of Hyperbolic Cosecant]]:
:$\dfrac \d {\d x} \csch x = -\csch x \coth x$
The result follows from the definition of [[Definition:Primitive (Calculus)|primitive]].
{{Qed}} | Primitive of Product of Hyperbolic Cosecant and Cotangent | https://proofwiki.org/wiki/Primitive_of_Product_of_Hyperbolic_Cosecant_and_Cotangent | https://proofwiki.org/wiki/Primitive_of_Product_of_Hyperbolic_Cosecant_and_Cotangent | [
"Primitives of Hyperbolic Functions",
"Hyperbolic Cosecant Function",
"Hyperbolic Cotangent Function"
] | [
"Definition:Primitive (Calculus)/Constant of Integration"
] | [
"Derivative of Hyperbolic Cosecant",
"Definition:Primitive (Calculus)"
] |
proofwiki-9072 | Primitive of Reciprocal of x squared minus a squared | Let $a \in \R_{>0}$ be a strictly positive real constant.
=== Inverse Hyperbolic Function Form ===
{{:Primitive of Reciprocal of x squared minus a squared/Inverse Hyperbolic Function Form}}
=== $1$st Logarithm Form ===
{{:Primitive of Reciprocal of x squared minus a squared/Logarithm Form 1}}
=== $2$nd Logarithm Form =... | Let $\size x > a$.
Let:
{{begin-eqn}}
{{eqn | l = u
| r = \coth^{-1} {\frac x a}
| c = {{Defof|Real Inverse Hyperbolic Cotangent}}, which is defined where $\size {\dfrac x a} > 1$
}}
{{eqn | ll= \leadsto
| l = x
| r = a \coth u
| c =
}}
{{eqn | ll= \leadsto
| l = \frac {\d x} {\d u}... | Let $a \in \R_{>0}$ be a [[Definition:Strictly Positive Real Number|strictly positive real]] [[Definition:Constant|constant]].
=== [[Primitive of Reciprocal of x squared minus a squared/Inverse Hyperbolic Function Form|Inverse Hyperbolic Function Form]] ===
{{:Primitive of Reciprocal of x squared minus a squared/Inve... | Let $\size x > a$.
Let:
{{begin-eqn}}
{{eqn | l = u
| r = \coth^{-1} {\frac x a}
| c = {{Defof|Real Inverse Hyperbolic Cotangent}}, which is defined where $\size {\dfrac x a} > 1$
}}
{{eqn | ll= \leadsto
| l = x
| r = a \coth u
| c =
}}
{{eqn | ll= \leadsto
| l = \frac {\d x} {\d ... | Primitive of Reciprocal of x squared minus a squared/Inverse Hyperbolic Cotangent Form/Proof | https://proofwiki.org/wiki/Primitive_of_Reciprocal_of_x_squared_minus_a_squared | https://proofwiki.org/wiki/Primitive_of_Reciprocal_of_x_squared_minus_a_squared/Inverse_Hyperbolic_Cotangent_Form/Proof | [
"Primitive of Reciprocal of x squared minus a squared",
"Primitives involving x squared minus a squared",
"Primitives involving Reciprocals"
] | [
"Definition:Strictly Positive/Real Number",
"Definition:Constant",
"Primitive of Reciprocal of x squared minus a squared/Inverse Hyperbolic Function Form",
"Primitive of Reciprocal of x squared minus a squared/Logarithm Form 1",
"Primitive of Reciprocal of x squared minus a squared/Logarithm Form 2"
] | [
"Derivative of Hyperbolic Cotangent",
"Integration by Substitution",
"Primitive of Constant Multiple of Function",
"Difference of Squares of Hyperbolic Cotangent and Cosecant",
"Integral of Constant"
] |
proofwiki-9073 | Primitive of Reciprocal of x squared minus a squared | Let $a \in \R_{>0}$ be a strictly positive real constant.
=== Inverse Hyperbolic Function Form ===
{{:Primitive of Reciprocal of x squared minus a squared/Inverse Hyperbolic Function Form}}
=== $1$st Logarithm Form ===
{{:Primitive of Reciprocal of x squared minus a squared/Logarithm Form 1}}
=== $2$nd Logarithm Form =... | Let $x > a$.
Then:
{{begin-eqn}}
{{eqn | l = \int \frac {\d x} {x^2 - a^2}
| r = -\frac 1 a \arcoth {\frac x a} + C
| c = Primitive of Reciprocal of $x^2 - a^2$ in $\arcoth$ form
}}
{{eqn | r = -\frac 1 a \paren {\dfrac 1 2 \map \ln {\dfrac {x + a} {x - a} } } + C
| c = $\arcoth {\dfrac x a}$ in Logar... | Let $a \in \R_{>0}$ be a [[Definition:Strictly Positive Real Number|strictly positive real]] [[Definition:Constant|constant]].
=== [[Primitive of Reciprocal of x squared minus a squared/Inverse Hyperbolic Function Form|Inverse Hyperbolic Function Form]] ===
{{:Primitive of Reciprocal of x squared minus a squared/Inve... | Let $x > a$.
Then:
{{begin-eqn}}
{{eqn | l = \int \frac {\d x} {x^2 - a^2}
| r = -\frac 1 a \arcoth {\frac x a} + C
| c = [[Primitive of Reciprocal of x squared minus a squared/Inverse Hyperbolic Cotangent Form|Primitive of Reciprocal of $x^2 - a^2$ in $\arcoth$ form]]
}}
{{eqn | r = -\frac 1 a \paren {\df... | Primitive of Reciprocal of x squared minus a squared/Logarithm Form 2/Proof 1 | https://proofwiki.org/wiki/Primitive_of_Reciprocal_of_x_squared_minus_a_squared | https://proofwiki.org/wiki/Primitive_of_Reciprocal_of_x_squared_minus_a_squared/Logarithm_Form_2/Proof_1 | [
"Primitive of Reciprocal of x squared minus a squared",
"Primitives involving x squared minus a squared",
"Primitives involving Reciprocals"
] | [
"Definition:Strictly Positive/Real Number",
"Definition:Constant",
"Primitive of Reciprocal of x squared minus a squared/Inverse Hyperbolic Function Form",
"Primitive of Reciprocal of x squared minus a squared/Logarithm Form 1",
"Primitive of Reciprocal of x squared minus a squared/Logarithm Form 2"
] | [
"Primitive of Reciprocal of x squared minus a squared/Inverse Hyperbolic Cotangent Form",
"Real Area Hyperbolic Cotangent of x over a in Logarithm Form",
"Logarithm of Reciprocal",
"Integration by Substitution",
"Logarithm of Reciprocal"
] |
proofwiki-9074 | Primitive of Reciprocal of x squared minus a squared | Let $a \in \R_{>0}$ be a strictly positive real constant.
=== Inverse Hyperbolic Function Form ===
{{:Primitive of Reciprocal of x squared minus a squared/Inverse Hyperbolic Function Form}}
=== $1$st Logarithm Form ===
{{:Primitive of Reciprocal of x squared minus a squared/Logarithm Form 1}}
=== $2$nd Logarithm Form =... | {{begin-eqn}}
{{eqn | l = \int \frac {\d x} {x^2 - a^2}
| r = \int \frac {\d x} {\paren {x - a} \paren {x + a} }
| c = Difference of Two Squares
}}
{{eqn | r = \int \frac {\d x} {2 a \paren {x - a} } - \int \frac {\d x} {2 a \paren {x + a} }
| c = Partial Fraction Expansion
}}
{{eqn | r = \frac 1 {2 a... | Let $a \in \R_{>0}$ be a [[Definition:Strictly Positive Real Number|strictly positive real]] [[Definition:Constant|constant]].
=== [[Primitive of Reciprocal of x squared minus a squared/Inverse Hyperbolic Function Form|Inverse Hyperbolic Function Form]] ===
{{:Primitive of Reciprocal of x squared minus a squared/Inve... | {{begin-eqn}}
{{eqn | l = \int \frac {\d x} {x^2 - a^2}
| r = \int \frac {\d x} {\paren {x - a} \paren {x + a} }
| c = [[Difference of Two Squares]]
}}
{{eqn | r = \int \frac {\d x} {2 a \paren {x - a} } - \int \frac {\d x} {2 a \paren {x + a} }
| c = [[Primitive of Reciprocal of x squared minus a squ... | Primitive of Reciprocal of x squared minus a squared/Logarithm Form 2/Proof 2 | https://proofwiki.org/wiki/Primitive_of_Reciprocal_of_x_squared_minus_a_squared | https://proofwiki.org/wiki/Primitive_of_Reciprocal_of_x_squared_minus_a_squared/Logarithm_Form_2/Proof_2 | [
"Primitive of Reciprocal of x squared minus a squared",
"Primitives involving x squared minus a squared",
"Primitives involving Reciprocals"
] | [
"Definition:Strictly Positive/Real Number",
"Definition:Constant",
"Primitive of Reciprocal of x squared minus a squared/Inverse Hyperbolic Function Form",
"Primitive of Reciprocal of x squared minus a squared/Logarithm Form 1",
"Primitive of Reciprocal of x squared minus a squared/Logarithm Form 2"
] | [
"Difference of Two Squares",
"Primitive of Reciprocal of x squared minus a squared/Logarithm Form 2/Proof 2/Partial Fraction Expansion",
"Primitive of Constant Multiple of Function",
"Primitive of Reciprocal",
"Difference of Logarithms"
] |
proofwiki-9075 | Primitive of Reciprocal of x squared minus a squared | Let $a \in \R_{>0}$ be a strictly positive real constant.
=== Inverse Hyperbolic Function Form ===
{{:Primitive of Reciprocal of x squared minus a squared/Inverse Hyperbolic Function Form}}
=== $1$st Logarithm Form ===
{{:Primitive of Reciprocal of x squared minus a squared/Logarithm Form 1}}
=== $2$nd Logarithm Form =... | From the $1$st logarithm form:
$\quad \ds \int \frac {\d x} {x^2 - a^2} = \begin {cases} \dfrac 1 {2 a} \map \ln {\dfrac {a - x} {a + x} } + C & : \size x < a \\
& \\
\dfrac 1 {2 a} \map \ln {\dfrac {x - a} {x + a} } + C & : \size x > a \\
& \\
\text {undefined} & : \size x = a \end {cases}$
From Primitive of Reciproca... | Let $a \in \R_{>0}$ be a [[Definition:Strictly Positive Real Number|strictly positive real]] [[Definition:Constant|constant]].
=== [[Primitive of Reciprocal of x squared minus a squared/Inverse Hyperbolic Function Form|Inverse Hyperbolic Function Form]] ===
{{:Primitive of Reciprocal of x squared minus a squared/Inve... | From the [[Primitive of Reciprocal of x squared minus a squared/Logarithm Form 1|$1$st logarithm form]]:
$\quad \ds \int \frac {\d x} {x^2 - a^2} = \begin {cases} \dfrac 1 {2 a} \map \ln {\dfrac {a - x} {a + x} } + C & : \size x < a \\
& \\
\dfrac 1 {2 a} \map \ln {\dfrac {x - a} {x + a} } + C & : \size x > a \\
& \\
... | Primitive of Reciprocal of x squared minus a squared/Logarithm Form 2/Proof 3 | https://proofwiki.org/wiki/Primitive_of_Reciprocal_of_x_squared_minus_a_squared | https://proofwiki.org/wiki/Primitive_of_Reciprocal_of_x_squared_minus_a_squared/Logarithm_Form_2/Proof_3 | [
"Primitive of Reciprocal of x squared minus a squared",
"Primitives involving x squared minus a squared",
"Primitives involving Reciprocals"
] | [
"Definition:Strictly Positive/Real Number",
"Definition:Constant",
"Primitive of Reciprocal of x squared minus a squared/Inverse Hyperbolic Function Form",
"Primitive of Reciprocal of x squared minus a squared/Logarithm Form 1",
"Primitive of Reciprocal of x squared minus a squared/Logarithm Form 2"
] | [
"Primitive of Reciprocal of x squared minus a squared/Logarithm Form 1",
"Primitive of Reciprocal of x squared minus a squared/Logarithm Form/Lemma"
] |
proofwiki-9076 | Primitive of Reciprocal of x squared minus a squared/Logarithm Form 2 | :$\ds \int \frac {\d x} {x^2 - a^2} = \frac 1 {2 a} \ln \size {\frac {x - a} {x + a} } + C$ | Let $x > a$.
Then:
{{begin-eqn}}
{{eqn | l = \int \frac {\d x} {x^2 - a^2}
| r = -\frac 1 a \arcoth {\frac x a} + C
| c = Primitive of Reciprocal of $x^2 - a^2$ in $\arcoth$ form
}}
{{eqn | r = -\frac 1 a \paren {\dfrac 1 2 \map \ln {\dfrac {x + a} {x - a} } } + C
| c = $\arcoth {\dfrac x a}$ in Logar... | :$\ds \int \frac {\d x} {x^2 - a^2} = \frac 1 {2 a} \ln \size {\frac {x - a} {x + a} } + C$ | Let $x > a$.
Then:
{{begin-eqn}}
{{eqn | l = \int \frac {\d x} {x^2 - a^2}
| r = -\frac 1 a \arcoth {\frac x a} + C
| c = [[Primitive of Reciprocal of x squared minus a squared/Inverse Hyperbolic Cotangent Form|Primitive of Reciprocal of $x^2 - a^2$ in $\arcoth$ form]]
}}
{{eqn | r = -\frac 1 a \paren {\df... | Primitive of Reciprocal of x squared minus a squared/Logarithm Form 2/Proof 1 | https://proofwiki.org/wiki/Primitive_of_Reciprocal_of_x_squared_minus_a_squared/Logarithm_Form_2 | https://proofwiki.org/wiki/Primitive_of_Reciprocal_of_x_squared_minus_a_squared/Logarithm_Form_2/Proof_1 | [
"Primitive of Reciprocal of x squared minus a squared"
] | [] | [
"Primitive of Reciprocal of x squared minus a squared/Inverse Hyperbolic Cotangent Form",
"Real Area Hyperbolic Cotangent of x over a in Logarithm Form",
"Logarithm of Reciprocal",
"Integration by Substitution",
"Logarithm of Reciprocal"
] |
proofwiki-9077 | Primitive of Reciprocal of x squared minus a squared/Logarithm Form 2 | :$\ds \int \frac {\d x} {x^2 - a^2} = \frac 1 {2 a} \ln \size {\frac {x - a} {x + a} } + C$ | {{begin-eqn}}
{{eqn | l = \int \frac {\d x} {x^2 - a^2}
| r = \int \frac {\d x} {\paren {x - a} \paren {x + a} }
| c = Difference of Two Squares
}}
{{eqn | r = \int \frac {\d x} {2 a \paren {x - a} } - \int \frac {\d x} {2 a \paren {x + a} }
| c = Partial Fraction Expansion
}}
{{eqn | r = \frac 1 {2 a... | :$\ds \int \frac {\d x} {x^2 - a^2} = \frac 1 {2 a} \ln \size {\frac {x - a} {x + a} } + C$ | {{begin-eqn}}
{{eqn | l = \int \frac {\d x} {x^2 - a^2}
| r = \int \frac {\d x} {\paren {x - a} \paren {x + a} }
| c = [[Difference of Two Squares]]
}}
{{eqn | r = \int \frac {\d x} {2 a \paren {x - a} } - \int \frac {\d x} {2 a \paren {x + a} }
| c = [[Primitive of Reciprocal of x squared minus a squ... | Primitive of Reciprocal of x squared minus a squared/Logarithm Form 2/Proof 2 | https://proofwiki.org/wiki/Primitive_of_Reciprocal_of_x_squared_minus_a_squared/Logarithm_Form_2 | https://proofwiki.org/wiki/Primitive_of_Reciprocal_of_x_squared_minus_a_squared/Logarithm_Form_2/Proof_2 | [
"Primitive of Reciprocal of x squared minus a squared"
] | [] | [
"Difference of Two Squares",
"Primitive of Reciprocal of x squared minus a squared/Logarithm Form 2/Proof 2/Partial Fraction Expansion",
"Primitive of Constant Multiple of Function",
"Primitive of Reciprocal",
"Difference of Logarithms"
] |
proofwiki-9078 | Primitive of Reciprocal of x squared minus a squared/Logarithm Form 2 | :$\ds \int \frac {\d x} {x^2 - a^2} = \frac 1 {2 a} \ln \size {\frac {x - a} {x + a} } + C$ | From the $1$st logarithm form:
$\quad \ds \int \frac {\d x} {x^2 - a^2} = \begin {cases} \dfrac 1 {2 a} \map \ln {\dfrac {a - x} {a + x} } + C & : \size x < a \\
& \\
\dfrac 1 {2 a} \map \ln {\dfrac {x - a} {x + a} } + C & : \size x > a \\
& \\
\text {undefined} & : \size x = a \end {cases}$
From Primitive of Reciproca... | :$\ds \int \frac {\d x} {x^2 - a^2} = \frac 1 {2 a} \ln \size {\frac {x - a} {x + a} } + C$ | From the [[Primitive of Reciprocal of x squared minus a squared/Logarithm Form 1|$1$st logarithm form]]:
$\quad \ds \int \frac {\d x} {x^2 - a^2} = \begin {cases} \dfrac 1 {2 a} \map \ln {\dfrac {a - x} {a + x} } + C & : \size x < a \\
& \\
\dfrac 1 {2 a} \map \ln {\dfrac {x - a} {x + a} } + C & : \size x > a \\
& \\
... | Primitive of Reciprocal of x squared minus a squared/Logarithm Form 2/Proof 3 | https://proofwiki.org/wiki/Primitive_of_Reciprocal_of_x_squared_minus_a_squared/Logarithm_Form_2 | https://proofwiki.org/wiki/Primitive_of_Reciprocal_of_x_squared_minus_a_squared/Logarithm_Form_2/Proof_3 | [
"Primitive of Reciprocal of x squared minus a squared"
] | [] | [
"Primitive of Reciprocal of x squared minus a squared/Logarithm Form 1",
"Primitive of Reciprocal of x squared minus a squared/Logarithm Form/Lemma"
] |
proofwiki-9079 | Primitive of Reciprocal of a squared minus x squared/Logarithm Form 1 | $\quad \ds \int \frac {\d x} {a^2 - x^2} = \begin {cases} \dfrac 1 {2 a} \map \ln {\dfrac {a + x} {a - x} } + C & : \size x < a \\
& \\
\dfrac 1 {2 a} \map \ln {\dfrac {x + a} {x - a} } + C & : \size x > a \\
& \\
\text {undefined} & : \size x = a \end {cases}$ | === Case where $\size x < a$ ===
{{:Primitive of Reciprocal of a squared minus x squared/Logarithm Form 1/size of x less than a}} | $\quad \ds \int \frac {\d x} {a^2 - x^2} = \begin {cases} \dfrac 1 {2 a} \map \ln {\dfrac {a + x} {a - x} } + C & : \size x < a \\
& \\
\dfrac 1 {2 a} \map \ln {\dfrac {x + a} {x - a} } + C & : \size x > a \\
& \\
\text {undefined} & : \size x = a \end {cases}$ | === [[Primitive of Reciprocal of a squared minus x squared/Logarithm Form 1/size of x less than a|Case where $\size x < a$]] ===
{{:Primitive of Reciprocal of a squared minus x squared/Logarithm Form 1/size of x less than a}} | Primitive of Reciprocal of a squared minus x squared/Logarithm Form 1 | https://proofwiki.org/wiki/Primitive_of_Reciprocal_of_a_squared_minus_x_squared/Logarithm_Form_1 | https://proofwiki.org/wiki/Primitive_of_Reciprocal_of_a_squared_minus_x_squared/Logarithm_Form_1 | [
"Primitive of Reciprocal of a squared minus x squared"
] | [] | [
"Primitive of Reciprocal of a squared minus x squared/Logarithm Form 1/size of x less than a"
] |
proofwiki-9080 | Primitive of Reciprocal of a squared minus x squared/Logarithm Form 1 | $\quad \ds \int \frac {\d x} {a^2 - x^2} = \begin {cases} \dfrac 1 {2 a} \map \ln {\dfrac {a + x} {a - x} } + C & : \size x < a \\
& \\
\dfrac 1 {2 a} \map \ln {\dfrac {x + a} {x - a} } + C & : \size x > a \\
& \\
\text {undefined} & : \size x = a \end {cases}$ | Let $\size x > a$.
Then:
{{begin-eqn}}
{{eqn | l = \int \frac {\d x} {a^2 - x^2}
| r = \frac 1 a \arcoth {\frac x a} + C
| c = Primitive of $\dfrac 1 {a^2 - x^2}$: $\arcoth$ form
}}
{{eqn | r = \frac 1 a \paren {\dfrac 1 2 \map \ln {\dfrac {x + a} {x - a} } } + C
| c = $\arcoth \dfrac x a$ in Logarit... | $\quad \ds \int \frac {\d x} {a^2 - x^2} = \begin {cases} \dfrac 1 {2 a} \map \ln {\dfrac {a + x} {a - x} } + C & : \size x < a \\
& \\
\dfrac 1 {2 a} \map \ln {\dfrac {x + a} {x - a} } + C & : \size x > a \\
& \\
\text {undefined} & : \size x = a \end {cases}$ | Let $\size x > a$.
Then:
{{begin-eqn}}
{{eqn | l = \int \frac {\d x} {a^2 - x^2}
| r = \frac 1 a \arcoth {\frac x a} + C
| c = [[Primitive of Reciprocal of a squared minus x squared/Inverse Hyperbolic Cotangent Form|Primitive of $\dfrac 1 {a^2 - x^2}$: $\arcoth$ form]]
}}
{{eqn | r = \frac 1 a \paren {\dfr... | Primitive of Reciprocal of a squared minus x squared/Logarithm Form 1/size of x greater than a/Proof 1 | https://proofwiki.org/wiki/Primitive_of_Reciprocal_of_a_squared_minus_x_squared/Logarithm_Form_1 | https://proofwiki.org/wiki/Primitive_of_Reciprocal_of_a_squared_minus_x_squared/Logarithm_Form_1/size_of_x_greater_than_a/Proof_1 | [
"Primitive of Reciprocal of a squared minus x squared"
] | [] | [
"Primitive of Reciprocal of a squared minus x squared/Inverse Hyperbolic Cotangent Form",
"Real Area Hyperbolic Cotangent of x over a in Logarithm Form"
] |
proofwiki-9081 | Primitive of Reciprocal of a squared minus x squared/Logarithm Form 1 | $\quad \ds \int \frac {\d x} {a^2 - x^2} = \begin {cases} \dfrac 1 {2 a} \map \ln {\dfrac {a + x} {a - x} } + C & : \size x < a \\
& \\
\dfrac 1 {2 a} \map \ln {\dfrac {x + a} {x - a} } + C & : \size x > a \\
& \\
\text {undefined} & : \size x = a \end {cases}$ | Let $\size x < a$.
Then:
{{begin-eqn}}
{{eqn | l = \int \frac {\d x} {a^2 - x^2}
| r = \frac 1 a \artanh {\frac x a} + C
| c = Primitive of $\dfrac 1 {a^2 - x^2}$: $\artanh$ form
}}
{{eqn | r = \frac 1 a \paren {\dfrac 1 2 \map \ln {\dfrac {a + x} {a - x} } } + C
| c = $\artanh \dfrac x a$ in Logarith... | $\quad \ds \int \frac {\d x} {a^2 - x^2} = \begin {cases} \dfrac 1 {2 a} \map \ln {\dfrac {a + x} {a - x} } + C & : \size x < a \\
& \\
\dfrac 1 {2 a} \map \ln {\dfrac {x + a} {x - a} } + C & : \size x > a \\
& \\
\text {undefined} & : \size x = a \end {cases}$ | Let $\size x < a$.
Then:
{{begin-eqn}}
{{eqn | l = \int \frac {\d x} {a^2 - x^2}
| r = \frac 1 a \artanh {\frac x a} + C
| c = [[Primitive of Reciprocal of a squared minus x squared/Inverse Hyperbolic Tangent Form|Primitive of $\dfrac 1 {a^2 - x^2}$: $\artanh$ form]]
}}
{{eqn | r = \frac 1 a \paren {\dfrac... | Primitive of Reciprocal of a squared minus x squared/Logarithm Form 1/size of x less than a/Proof 1 | https://proofwiki.org/wiki/Primitive_of_Reciprocal_of_a_squared_minus_x_squared/Logarithm_Form_1 | https://proofwiki.org/wiki/Primitive_of_Reciprocal_of_a_squared_minus_x_squared/Logarithm_Form_1/size_of_x_less_than_a/Proof_1 | [
"Primitive of Reciprocal of a squared minus x squared"
] | [] | [
"Primitive of Reciprocal of a squared minus x squared/Inverse Hyperbolic Tangent Form",
"Real Area Hyperbolic Tangent of x over a in Logarithm Form"
] |
proofwiki-9082 | Primitive of Reciprocal of a squared minus x squared/Logarithm Form 1 | $\quad \ds \int \frac {\d x} {a^2 - x^2} = \begin {cases} \dfrac 1 {2 a} \map \ln {\dfrac {a + x} {a - x} } + C & : \size x < a \\
& \\
\dfrac 1 {2 a} \map \ln {\dfrac {x + a} {x - a} } + C & : \size x > a \\
& \\
\text {undefined} & : \size x = a \end {cases}$ | Let $\size x < a$.
Then:
{{begin-eqn}}
{{eqn | l = \int \frac {\d x} {a^2 - x^2}
| r = \int \frac {\d x} {\paren {a + x} \paren {a - x} }
| c = Difference of Two Squares
}}
{{eqn | r = \int \frac {\d x} {2 a \paren {a + x} } + \int \frac {\d x} {2 a \paren {a - x} }
| c = Partial Fraction Expansion
}}... | $\quad \ds \int \frac {\d x} {a^2 - x^2} = \begin {cases} \dfrac 1 {2 a} \map \ln {\dfrac {a + x} {a - x} } + C & : \size x < a \\
& \\
\dfrac 1 {2 a} \map \ln {\dfrac {x + a} {x - a} } + C & : \size x > a \\
& \\
\text {undefined} & : \size x = a \end {cases}$ | Let $\size x < a$.
Then:
{{begin-eqn}}
{{eqn | l = \int \frac {\d x} {a^2 - x^2}
| r = \int \frac {\d x} {\paren {a + x} \paren {a - x} }
| c = [[Difference of Two Squares]]
}}
{{eqn | r = \int \frac {\d x} {2 a \paren {a + x} } + \int \frac {\d x} {2 a \paren {a - x} }
| c = [[Primitive of Reciproca... | Primitive of Reciprocal of a squared minus x squared/Logarithm Form 1/size of x less than a/Proof 2 | https://proofwiki.org/wiki/Primitive_of_Reciprocal_of_a_squared_minus_x_squared/Logarithm_Form_1 | https://proofwiki.org/wiki/Primitive_of_Reciprocal_of_a_squared_minus_x_squared/Logarithm_Form_1/size_of_x_less_than_a/Proof_2 | [
"Primitive of Reciprocal of a squared minus x squared"
] | [] | [
"Difference of Two Squares",
"Primitive of Reciprocal of a squared minus x squared/Logarithm Form/Partial Fraction Expansion",
"Primitive of Constant Multiple of Function",
"Primitive of Reciprocal",
"Difference of Logarithms"
] |
proofwiki-9083 | Primitive of Reciprocal of a squared minus x squared/Logarithm Form 1 | $\quad \ds \int \frac {\d x} {a^2 - x^2} = \begin {cases} \dfrac 1 {2 a} \map \ln {\dfrac {a + x} {a - x} } + C & : \size x < a \\
& \\
\dfrac 1 {2 a} \map \ln {\dfrac {x + a} {x - a} } + C & : \size x > a \\
& \\
\text {undefined} & : \size x = a \end {cases}$ | Let $\size x < a$.
Then:
{{begin-eqn}}
{{eqn | l = \int \frac {\d x} {a^2 - x^2}
| r = -\int \frac {\d x} {x^2 - a^2}
| c = Primitive of Constant Multiple of Function
}}
{{eqn | r = -\frac 1 {2 a} \map \ln {\frac {a - x} {a + x} } + C
| c = Primitive of $\dfrac 1 {x^2 - a^2}$ for $\size x < a$
}}
{{eq... | $\quad \ds \int \frac {\d x} {a^2 - x^2} = \begin {cases} \dfrac 1 {2 a} \map \ln {\dfrac {a + x} {a - x} } + C & : \size x < a \\
& \\
\dfrac 1 {2 a} \map \ln {\dfrac {x + a} {x - a} } + C & : \size x > a \\
& \\
\text {undefined} & : \size x = a \end {cases}$ | Let $\size x < a$.
Then:
{{begin-eqn}}
{{eqn | l = \int \frac {\d x} {a^2 - x^2}
| r = -\int \frac {\d x} {x^2 - a^2}
| c = [[Primitive of Constant Multiple of Function]]
}}
{{eqn | r = -\frac 1 {2 a} \map \ln {\frac {a - x} {a + x} } + C
| c = [[Primitive of Reciprocal of x squared minus a squared/L... | Primitive of Reciprocal of a squared minus x squared/Logarithm Form 1/size of x less than a/Proof 3 | https://proofwiki.org/wiki/Primitive_of_Reciprocal_of_a_squared_minus_x_squared/Logarithm_Form_1 | https://proofwiki.org/wiki/Primitive_of_Reciprocal_of_a_squared_minus_x_squared/Logarithm_Form_1/size_of_x_less_than_a/Proof_3 | [
"Primitive of Reciprocal of a squared minus x squared"
] | [] | [
"Primitive of Constant Multiple of Function",
"Primitive of Reciprocal of x squared minus a squared/Logarithm Form 1/size of x less than a",
"Logarithm of Reciprocal"
] |
proofwiki-9084 | Primitive of Reciprocal of a squared minus x squared | Let $a \in \R_{>0}$ be a strictly positive real constant.
=== Inverse Hyperbolic Function Form ===
{{:Primitive of Reciprocal of a squared minus x squared/Inverse Hyperbolic Function Form}}
=== $1$st Logarithm Form ===
{{:Primitive of Reciprocal of a squared minus x squared/Logarithm Form 1}}
=== $2$nd Logarithm Form =... | Let $\size x > a$.
Let:
{{begin-eqn}}
{{eqn | l = u
| r = \coth^{-1} {\frac x a}
| c = {{Defof|Real Inverse Hyperbolic Cotangent}}, which is defined where $\dfrac x a > 1$
}}
{{eqn | ll=\leadsto
| l = x
| r = a \coth u
| c =
}}
{{eqn | ll=\leadsto
| l = \frac {\d x} {\d u}
| r... | Let $a \in \R_{>0}$ be a [[Definition:Strictly Positive Real Number|strictly positive real]] [[Definition:Constant|constant]].
=== [[Primitive of Reciprocal of a squared minus x squared/Inverse Hyperbolic Function Form|Inverse Hyperbolic Function Form]] ===
{{:Primitive of Reciprocal of a squared minus x squared/Inve... | Let $\size x > a$.
Let:
{{begin-eqn}}
{{eqn | l = u
| r = \coth^{-1} {\frac x a}
| c = {{Defof|Real Inverse Hyperbolic Cotangent}}, which is defined where $\dfrac x a > 1$
}}
{{eqn | ll=\leadsto
| l = x
| r = a \coth u
| c =
}}
{{eqn | ll=\leadsto
| l = \frac {\d x} {\d u}
... | Primitive of Reciprocal of a squared minus x squared/Inverse Hyperbolic Cotangent Form/Proof | https://proofwiki.org/wiki/Primitive_of_Reciprocal_of_a_squared_minus_x_squared | https://proofwiki.org/wiki/Primitive_of_Reciprocal_of_a_squared_minus_x_squared/Inverse_Hyperbolic_Cotangent_Form/Proof | [
"Primitive of Reciprocal of a squared minus x squared",
"Primitives involving a squared minus x squared",
"Primitives involving Reciprocals"
] | [
"Definition:Strictly Positive/Real Number",
"Definition:Constant",
"Primitive of Reciprocal of a squared minus x squared/Inverse Hyperbolic Function Form",
"Primitive of Reciprocal of a squared minus x squared/Logarithm Form 1",
"Primitive of Reciprocal of a squared minus x squared/Logarithm Form 2"
] | [
"Derivative of Hyperbolic Cotangent",
"Integration by Substitution",
"Primitive of Constant Multiple of Function",
"Difference of Squares of Hyperbolic Cotangent and Cosecant",
"Integral of Constant"
] |
proofwiki-9085 | Primitive of Reciprocal of a squared minus x squared | Let $a \in \R_{>0}$ be a strictly positive real constant.
=== Inverse Hyperbolic Function Form ===
{{:Primitive of Reciprocal of a squared minus x squared/Inverse Hyperbolic Function Form}}
=== $1$st Logarithm Form ===
{{:Primitive of Reciprocal of a squared minus x squared/Logarithm Form 1}}
=== $2$nd Logarithm Form =... | Let $\size x < a$.
Let:
{{begin-eqn}}
{{eqn | l = u
| r = \tanh^{-1} {\frac x a}
| c = {{Defof|Real Inverse Hyperbolic Tangent}}, which is defined where $\size {\dfrac x a} < 1$
}}
{{eqn | ll=\leadsto
| l = x
| r = a \tanh u
| c =
}}
{{eqn | ll=\leadsto
| l = \frac {\d x} {\d u}
... | Let $a \in \R_{>0}$ be a [[Definition:Strictly Positive Real Number|strictly positive real]] [[Definition:Constant|constant]].
=== [[Primitive of Reciprocal of a squared minus x squared/Inverse Hyperbolic Function Form|Inverse Hyperbolic Function Form]] ===
{{:Primitive of Reciprocal of a squared minus x squared/Inve... | Let $\size x < a$.
Let:
{{begin-eqn}}
{{eqn | l = u
| r = \tanh^{-1} {\frac x a}
| c = {{Defof|Real Inverse Hyperbolic Tangent}}, which is defined where $\size {\dfrac x a} < 1$
}}
{{eqn | ll=\leadsto
| l = x
| r = a \tanh u
| c =
}}
{{eqn | ll=\leadsto
| l = \frac {\d x} {\d u}
... | Primitive of Reciprocal of a squared minus x squared/Inverse Hyperbolic Tangent Form/Proof | https://proofwiki.org/wiki/Primitive_of_Reciprocal_of_a_squared_minus_x_squared | https://proofwiki.org/wiki/Primitive_of_Reciprocal_of_a_squared_minus_x_squared/Inverse_Hyperbolic_Tangent_Form/Proof | [
"Primitive of Reciprocal of a squared minus x squared",
"Primitives involving a squared minus x squared",
"Primitives involving Reciprocals"
] | [
"Definition:Strictly Positive/Real Number",
"Definition:Constant",
"Primitive of Reciprocal of a squared minus x squared/Inverse Hyperbolic Function Form",
"Primitive of Reciprocal of a squared minus x squared/Logarithm Form 1",
"Primitive of Reciprocal of a squared minus x squared/Logarithm Form 2"
] | [
"Derivative of Hyperbolic Tangent",
"Integration by Substitution",
"Primitive of Constant Multiple of Function",
"Sum of Squares of Hyperbolic Secant and Tangent",
"Integral of Constant"
] |
proofwiki-9086 | Primitive of Reciprocal of a squared minus x squared | Let $a \in \R_{>0}$ be a strictly positive real constant.
=== Inverse Hyperbolic Function Form ===
{{:Primitive of Reciprocal of a squared minus x squared/Inverse Hyperbolic Function Form}}
=== $1$st Logarithm Form ===
{{:Primitive of Reciprocal of a squared minus x squared/Logarithm Form 1}}
=== $2$nd Logarithm Form =... | Let $\size x > a$.
Then:
{{begin-eqn}}
{{eqn | l = \int \frac {\d x} {a^2 - x^2}
| r = \frac 1 a \arcoth {\frac x a} + C
| c = Primitive of $\dfrac 1 {a^2 - x^2}$: $\arcoth$ form
}}
{{eqn | r = \frac 1 a \paren {\dfrac 1 2 \map \ln {\dfrac {x + a} {x - a} } } + C
| c = $\arcoth \dfrac x a$ in Logarit... | Let $a \in \R_{>0}$ be a [[Definition:Strictly Positive Real Number|strictly positive real]] [[Definition:Constant|constant]].
=== [[Primitive of Reciprocal of a squared minus x squared/Inverse Hyperbolic Function Form|Inverse Hyperbolic Function Form]] ===
{{:Primitive of Reciprocal of a squared minus x squared/Inve... | Let $\size x > a$.
Then:
{{begin-eqn}}
{{eqn | l = \int \frac {\d x} {a^2 - x^2}
| r = \frac 1 a \arcoth {\frac x a} + C
| c = [[Primitive of Reciprocal of a squared minus x squared/Inverse Hyperbolic Cotangent Form|Primitive of $\dfrac 1 {a^2 - x^2}$: $\arcoth$ form]]
}}
{{eqn | r = \frac 1 a \paren {\dfr... | Primitive of Reciprocal of a squared minus x squared/Logarithm Form 1/size of x greater than a/Proof 1 | https://proofwiki.org/wiki/Primitive_of_Reciprocal_of_a_squared_minus_x_squared | https://proofwiki.org/wiki/Primitive_of_Reciprocal_of_a_squared_minus_x_squared/Logarithm_Form_1/size_of_x_greater_than_a/Proof_1 | [
"Primitive of Reciprocal of a squared minus x squared",
"Primitives involving a squared minus x squared",
"Primitives involving Reciprocals"
] | [
"Definition:Strictly Positive/Real Number",
"Definition:Constant",
"Primitive of Reciprocal of a squared minus x squared/Inverse Hyperbolic Function Form",
"Primitive of Reciprocal of a squared minus x squared/Logarithm Form 1",
"Primitive of Reciprocal of a squared minus x squared/Logarithm Form 2"
] | [
"Primitive of Reciprocal of a squared minus x squared/Inverse Hyperbolic Cotangent Form",
"Real Area Hyperbolic Cotangent of x over a in Logarithm Form"
] |
proofwiki-9087 | Primitive of Reciprocal of a squared minus x squared | Let $a \in \R_{>0}$ be a strictly positive real constant.
=== Inverse Hyperbolic Function Form ===
{{:Primitive of Reciprocal of a squared minus x squared/Inverse Hyperbolic Function Form}}
=== $1$st Logarithm Form ===
{{:Primitive of Reciprocal of a squared minus x squared/Logarithm Form 1}}
=== $2$nd Logarithm Form =... | Let $\size x < a$.
Then:
{{begin-eqn}}
{{eqn | l = \int \frac {\d x} {a^2 - x^2}
| r = \frac 1 a \artanh {\frac x a} + C
| c = Primitive of $\dfrac 1 {a^2 - x^2}$: $\artanh$ form
}}
{{eqn | r = \frac 1 a \paren {\dfrac 1 2 \map \ln {\dfrac {a + x} {a - x} } } + C
| c = $\artanh \dfrac x a$ in Logarith... | Let $a \in \R_{>0}$ be a [[Definition:Strictly Positive Real Number|strictly positive real]] [[Definition:Constant|constant]].
=== [[Primitive of Reciprocal of a squared minus x squared/Inverse Hyperbolic Function Form|Inverse Hyperbolic Function Form]] ===
{{:Primitive of Reciprocal of a squared minus x squared/Inve... | Let $\size x < a$.
Then:
{{begin-eqn}}
{{eqn | l = \int \frac {\d x} {a^2 - x^2}
| r = \frac 1 a \artanh {\frac x a} + C
| c = [[Primitive of Reciprocal of a squared minus x squared/Inverse Hyperbolic Tangent Form|Primitive of $\dfrac 1 {a^2 - x^2}$: $\artanh$ form]]
}}
{{eqn | r = \frac 1 a \paren {\dfrac... | Primitive of Reciprocal of a squared minus x squared/Logarithm Form 1/size of x less than a/Proof 1 | https://proofwiki.org/wiki/Primitive_of_Reciprocal_of_a_squared_minus_x_squared | https://proofwiki.org/wiki/Primitive_of_Reciprocal_of_a_squared_minus_x_squared/Logarithm_Form_1/size_of_x_less_than_a/Proof_1 | [
"Primitive of Reciprocal of a squared minus x squared",
"Primitives involving a squared minus x squared",
"Primitives involving Reciprocals"
] | [
"Definition:Strictly Positive/Real Number",
"Definition:Constant",
"Primitive of Reciprocal of a squared minus x squared/Inverse Hyperbolic Function Form",
"Primitive of Reciprocal of a squared minus x squared/Logarithm Form 1",
"Primitive of Reciprocal of a squared minus x squared/Logarithm Form 2"
] | [
"Primitive of Reciprocal of a squared minus x squared/Inverse Hyperbolic Tangent Form",
"Real Area Hyperbolic Tangent of x over a in Logarithm Form"
] |
proofwiki-9088 | Primitive of Reciprocal of a squared minus x squared | Let $a \in \R_{>0}$ be a strictly positive real constant.
=== Inverse Hyperbolic Function Form ===
{{:Primitive of Reciprocal of a squared minus x squared/Inverse Hyperbolic Function Form}}
=== $1$st Logarithm Form ===
{{:Primitive of Reciprocal of a squared minus x squared/Logarithm Form 1}}
=== $2$nd Logarithm Form =... | Let $\size x < a$.
Then:
{{begin-eqn}}
{{eqn | l = \int \frac {\d x} {a^2 - x^2}
| r = \int \frac {\d x} {\paren {a + x} \paren {a - x} }
| c = Difference of Two Squares
}}
{{eqn | r = \int \frac {\d x} {2 a \paren {a + x} } + \int \frac {\d x} {2 a \paren {a - x} }
| c = Partial Fraction Expansion
}}... | Let $a \in \R_{>0}$ be a [[Definition:Strictly Positive Real Number|strictly positive real]] [[Definition:Constant|constant]].
=== [[Primitive of Reciprocal of a squared minus x squared/Inverse Hyperbolic Function Form|Inverse Hyperbolic Function Form]] ===
{{:Primitive of Reciprocal of a squared minus x squared/Inve... | Let $\size x < a$.
Then:
{{begin-eqn}}
{{eqn | l = \int \frac {\d x} {a^2 - x^2}
| r = \int \frac {\d x} {\paren {a + x} \paren {a - x} }
| c = [[Difference of Two Squares]]
}}
{{eqn | r = \int \frac {\d x} {2 a \paren {a + x} } + \int \frac {\d x} {2 a \paren {a - x} }
| c = [[Primitive of Reciproca... | Primitive of Reciprocal of a squared minus x squared/Logarithm Form 1/size of x less than a/Proof 2 | https://proofwiki.org/wiki/Primitive_of_Reciprocal_of_a_squared_minus_x_squared | https://proofwiki.org/wiki/Primitive_of_Reciprocal_of_a_squared_minus_x_squared/Logarithm_Form_1/size_of_x_less_than_a/Proof_2 | [
"Primitive of Reciprocal of a squared minus x squared",
"Primitives involving a squared minus x squared",
"Primitives involving Reciprocals"
] | [
"Definition:Strictly Positive/Real Number",
"Definition:Constant",
"Primitive of Reciprocal of a squared minus x squared/Inverse Hyperbolic Function Form",
"Primitive of Reciprocal of a squared minus x squared/Logarithm Form 1",
"Primitive of Reciprocal of a squared minus x squared/Logarithm Form 2"
] | [
"Difference of Two Squares",
"Primitive of Reciprocal of a squared minus x squared/Logarithm Form/Partial Fraction Expansion",
"Primitive of Constant Multiple of Function",
"Primitive of Reciprocal",
"Difference of Logarithms"
] |
proofwiki-9089 | Primitive of Reciprocal of a squared minus x squared | Let $a \in \R_{>0}$ be a strictly positive real constant.
=== Inverse Hyperbolic Function Form ===
{{:Primitive of Reciprocal of a squared minus x squared/Inverse Hyperbolic Function Form}}
=== $1$st Logarithm Form ===
{{:Primitive of Reciprocal of a squared minus x squared/Logarithm Form 1}}
=== $2$nd Logarithm Form =... | Let $\size x < a$.
Then:
{{begin-eqn}}
{{eqn | l = \int \frac {\d x} {a^2 - x^2}
| r = -\int \frac {\d x} {x^2 - a^2}
| c = Primitive of Constant Multiple of Function
}}
{{eqn | r = -\frac 1 {2 a} \map \ln {\frac {a - x} {a + x} } + C
| c = Primitive of $\dfrac 1 {x^2 - a^2}$ for $\size x < a$
}}
{{eq... | Let $a \in \R_{>0}$ be a [[Definition:Strictly Positive Real Number|strictly positive real]] [[Definition:Constant|constant]].
=== [[Primitive of Reciprocal of a squared minus x squared/Inverse Hyperbolic Function Form|Inverse Hyperbolic Function Form]] ===
{{:Primitive of Reciprocal of a squared minus x squared/Inve... | Let $\size x < a$.
Then:
{{begin-eqn}}
{{eqn | l = \int \frac {\d x} {a^2 - x^2}
| r = -\int \frac {\d x} {x^2 - a^2}
| c = [[Primitive of Constant Multiple of Function]]
}}
{{eqn | r = -\frac 1 {2 a} \map \ln {\frac {a - x} {a + x} } + C
| c = [[Primitive of Reciprocal of x squared minus a squared/L... | Primitive of Reciprocal of a squared minus x squared/Logarithm Form 1/size of x less than a/Proof 3 | https://proofwiki.org/wiki/Primitive_of_Reciprocal_of_a_squared_minus_x_squared | https://proofwiki.org/wiki/Primitive_of_Reciprocal_of_a_squared_minus_x_squared/Logarithm_Form_1/size_of_x_less_than_a/Proof_3 | [
"Primitive of Reciprocal of a squared minus x squared",
"Primitives involving a squared minus x squared",
"Primitives involving Reciprocals"
] | [
"Definition:Strictly Positive/Real Number",
"Definition:Constant",
"Primitive of Reciprocal of a squared minus x squared/Inverse Hyperbolic Function Form",
"Primitive of Reciprocal of a squared minus x squared/Logarithm Form 1",
"Primitive of Reciprocal of a squared minus x squared/Logarithm Form 2"
] | [
"Primitive of Constant Multiple of Function",
"Primitive of Reciprocal of x squared minus a squared/Logarithm Form 1/size of x less than a",
"Logarithm of Reciprocal"
] |
proofwiki-9090 | Primitive of Reciprocal of Root of x squared plus a squared/Logarithm Form | :$\ds \int \frac {\d x} {\sqrt {x^2 + a^2} } = \map \ln {x + \sqrt {x^2 + a^2} } + C$ | {{begin-eqn}}
{{eqn | l = \int \frac {\d x} {\sqrt {x^2 + a^2} }
| r = \arsinh {\frac x a} + C
| c = Primitive of $\dfrac 1 {\sqrt {x^2 + a^2} }$ in $\arsinh$ form
}}
{{eqn | r = \map \ln {x + \sqrt {x^2 + a^2} } - \ln a + C
| c = $\arsinh \dfrac x a$ in Logarithm Form
}}
{{eqn | r = \map \ln {x + \sq... | :$\ds \int \frac {\d x} {\sqrt {x^2 + a^2} } = \map \ln {x + \sqrt {x^2 + a^2} } + C$ | {{begin-eqn}}
{{eqn | l = \int \frac {\d x} {\sqrt {x^2 + a^2} }
| r = \arsinh {\frac x a} + C
| c = [[Primitive of Reciprocal of Root of x squared plus a squared/Inverse Hyperbolic Sine Form|Primitive of $\dfrac 1 {\sqrt {x^2 + a^2} }$ in $\arsinh$ form]]
}}
{{eqn | r = \map \ln {x + \sqrt {x^2 + a^2} } - ... | Primitive of Reciprocal of Root of x squared plus a squared/Logarithm Form/Proof 1 | https://proofwiki.org/wiki/Primitive_of_Reciprocal_of_Root_of_x_squared_plus_a_squared/Logarithm_Form | https://proofwiki.org/wiki/Primitive_of_Reciprocal_of_Root_of_x_squared_plus_a_squared/Logarithm_Form/Proof_1 | [
"Primitive of Reciprocal of Root of x squared plus a squared"
] | [] | [
"Primitive of Reciprocal of Root of x squared plus a squared/Inverse Hyperbolic Sine Form",
"Real Area Hyperbolic Sine of x over a in Logarithm Form",
"Definition:Primitive (Calculus)/Constant of Integration"
] |
proofwiki-9091 | Primitive of Reciprocal of Root of x squared plus a squared/Logarithm Form | :$\ds \int \frac {\d x} {\sqrt {x^2 + a^2} } = \map \ln {x + \sqrt {x^2 + a^2} } + C$ | Let $y^2 = a^2 + x^2$.
Then:
{{begin-eqn}}
{{eqn | l = 2 y \frac {\d y} {\d x}
| r = 2 x
| c = Power Rule for Derivatives, Chain Rule for Derivatives
}}
{{eqn | ll= \leadsto
| l = y \frac {\d y} {\d x}
| r = x
| c = simplification
}}
{{eqn | ll= \leadsto
| l = \frac {\d y} x
| ... | :$\ds \int \frac {\d x} {\sqrt {x^2 + a^2} } = \map \ln {x + \sqrt {x^2 + a^2} } + C$ | Let $y^2 = a^2 + x^2$.
Then:
{{begin-eqn}}
{{eqn | l = 2 y \frac {\d y} {\d x}
| r = 2 x
| c = [[Power Rule for Derivatives]], [[Chain Rule for Derivatives]]
}}
{{eqn | ll= \leadsto
| l = y \frac {\d y} {\d x}
| r = x
| c = simplification
}}
{{eqn | ll= \leadsto
| l = \frac {\d y} ... | Primitive of Reciprocal of Root of x squared plus a squared/Logarithm Form/Proof 2 | https://proofwiki.org/wiki/Primitive_of_Reciprocal_of_Root_of_x_squared_plus_a_squared/Logarithm_Form | https://proofwiki.org/wiki/Primitive_of_Reciprocal_of_Root_of_x_squared_plus_a_squared/Logarithm_Form/Proof_2 | [
"Primitive of Reciprocal of Root of x squared plus a squared"
] | [] | [
"Power Rule for Derivatives",
"Derivative of Composite Function",
"Primitive of Function under its Derivative"
] |
proofwiki-9092 | Limit at Infinity of Polynomial over Complex Exponential | Let $n \in \N$.
Let $\map {P_n} x$ be a real polynomial, of degree $n$.
Let $e^z$ be the complex exponential function, where $z = x + i y$.
Let $a \in \R_{>0}$.
Then:
:$\ds \lim_{x \mathop \to +\infty} \frac {\map {P_n} x} {e^{a z} } = 0$ | Let $\epsilon > 0$.
By the definition of limits at infinity, we need to show that there is some $M \in \R$ such that:
:$\ds x > M \implies \size {\frac {\map {P_n} x} {e^{a z} } - 0} < \epsilon$
But:
{{begin-eqn}}
{{eqn | l = \size {\frac {\map {P_n} x}{e^{a z} } - 0}
| r = \frac {\size {\map {P_n} x} } {\size {e... | Let $n \in \N$.
Let $\map {P_n} x$ be a [[Definition:Real Polynomial Function|real polynomial]], of [[Definition:Degree of Polynomial|degree]] $n$.
Let $e^z$ be the [[Definition:Complex Exponential Function|complex exponential function]], where $z = x + i y$.
Let $a \in \R_{>0}$.
Then:
:$\ds \lim_{x \mathop \to +... | Let $\epsilon > 0$.
By the definition of [[Definition:Limit at Infinity|limits at infinity]], we need to show that there is some $M \in \R$ such that:
:$\ds x > M \implies \size {\frac {\map {P_n} x} {e^{a z} } - 0} < \epsilon$
But:
{{begin-eqn}}
{{eqn | l = \size {\frac {\map {P_n} x}{e^{a z} } - 0}
| r = \f... | Limit at Infinity of Polynomial over Complex Exponential | https://proofwiki.org/wiki/Limit_at_Infinity_of_Polynomial_over_Complex_Exponential | https://proofwiki.org/wiki/Limit_at_Infinity_of_Polynomial_over_Complex_Exponential | [
"Limits of Mappings",
"Polynomial Theory",
"Exponential Function"
] | [
"Definition:Polynomial Function/Real",
"Definition:Degree of Polynomial",
"Definition:Exponential Function/Complex"
] | [
"Definition:Limit of Real Function/Limit at Infinity/Positive",
"Complex Modulus of Product of Complex Numbers",
"Modulus of Exponential is Exponential of Real Part",
"Exponential Dominates Polynomial",
"Exponential Dominates Polynomial",
"Category:Limits of Mappings",
"Category:Polynomial Theory",
"C... |
proofwiki-9093 | Primitive of Reciprocal of Root of x squared minus a squared/Logarithm Form | :$\ds \int \frac {\d x} {\sqrt {x^2 - a^2} } = \ln \size {x + \sqrt {x^2 - a^2} } + C$
for $0 < a < \size x$. | We have that $\sqrt {x^2 - a^2}$ is defined only when $x^2 > a^2$, that is, either:
:$x > a$
or:
:$x < -a$
where it is assumed that $a > 0$.
Consider the arcsecant substitution:
:$u = \arcsec {\dfrac x a}$
which is defined for all $x$ such that $\size {\dfrac x a} \ge 1$.
That is:
:$\size x \ge a$
Hence from Shape of S... | :$\ds \int \frac {\d x} {\sqrt {x^2 - a^2} } = \ln \size {x + \sqrt {x^2 - a^2} } + C$
for $0 < a < \size x$. | We have that $\sqrt {x^2 - a^2}$ is defined only when $x^2 > a^2$, that is, either:
:$x > a$
or:
:$x < -a$
where it is assumed that $a > 0$.
Consider the [[Definition:Arcsecant|arcsecant]] substitution:
:$u = \arcsec {\dfrac x a}$
which is defined for all $x$ such that $\size {\dfrac x a} \ge 1$.
That is:
:$\size... | Primitive of Reciprocal of Root of x squared minus a squared/Logarithm Form | https://proofwiki.org/wiki/Primitive_of_Reciprocal_of_Root_of_x_squared_minus_a_squared/Logarithm_Form | https://proofwiki.org/wiki/Primitive_of_Reciprocal_of_Root_of_x_squared_minus_a_squared/Logarithm_Form | [
"Primitive of Reciprocal of Root of x squared minus a squared"
] | [] | [
"Definition:Inverse Secant/Real/Arcsecant",
"Shape of Secant Function",
"Derivative of Secant Function",
"Integration by Substitution",
"Sum of Squares of Sine and Cosine/Corollary 1",
"Primitive of Secant Function/Secant plus Tangent Form",
"Sum of Squares of Sine and Cosine/Corollary 1",
"Difference... |
proofwiki-9094 | Primitive of Reciprocal of Root of x squared minus a squared/Logarithm Form | :$\ds \int \frac {\d x} {\sqrt {x^2 - a^2} } = \ln \size {x + \sqrt {x^2 - a^2} } + C$
for $0 < a < \size x$. | {{begin-eqn}}
{{eqn | l = \int \frac {\d x} {\sqrt {x^2 - a^2} }
| r = \cosh^{-1} {\frac x a} + C'
| c = Primitive of Reciprocal of $\sqrt {x^2 - a^2}$: $\cosh^{-1}$ form
}}
{{eqn | r = \map \ln {\frac x a + \sqrt {\paren {\frac x a}^2 - 1} } + C'
| c = {{Defof|Real Inverse Hyperbolic Cosine}}
}}
{{eq... | :$\ds \int \frac {\d x} {\sqrt {x^2 - a^2} } = \ln \size {x + \sqrt {x^2 - a^2} } + C$
for $0 < a < \size x$. | {{begin-eqn}}
{{eqn | l = \int \frac {\d x} {\sqrt {x^2 - a^2} }
| r = \cosh^{-1} {\frac x a} + C'
| c = [[Primitive of Reciprocal of Root of x squared minus a squared/Inverse Hyperbolic Cosine Form|Primitive of Reciprocal of $\sqrt {x^2 - a^2}$: $\cosh^{-1}$ form]]
}}
{{eqn | r = \map \ln {\frac x a + \sqr... | Primitive of Reciprocal of Root of x squared minus a squared/Logarithm Form/Proof 1 | https://proofwiki.org/wiki/Primitive_of_Reciprocal_of_Root_of_x_squared_minus_a_squared/Logarithm_Form | https://proofwiki.org/wiki/Primitive_of_Reciprocal_of_Root_of_x_squared_minus_a_squared/Logarithm_Form/Proof_1 | [
"Primitive of Reciprocal of Root of x squared minus a squared"
] | [] | [
"Primitive of Reciprocal of Root of x squared minus a squared/Inverse Hyperbolic Cosine Form",
"Difference of Logarithms"
] |
proofwiki-9095 | Primitive of Reciprocal of x by Root of x squared minus a squared/Arcsecant Form | :$\ds \int \frac {\d x} {x \sqrt {x^2 - a^2} } = \frac 1 a \arcsec \size {\frac x a} + C$
for $0 < a < \size x$. | We have that $\sqrt {x^2 - a^2}$ is defined only when $x^2 > a^2$, that is, either:
:$x > a$
or:
:$x < -a$
where it is assumed that $a > 0$.
Consider the arcsecant substitution:
:$u = \arcsec {\dfrac x a}$
which is defined for all $x$ such that $\size {\dfrac x a} \ge 1$.
That is:
:$\size x \ge a$
and it is seen that $... | :$\ds \int \frac {\d x} {x \sqrt {x^2 - a^2} } = \frac 1 a \arcsec \size {\frac x a} + C$
for $0 < a < \size x$. | We have that $\sqrt {x^2 - a^2}$ is defined only when $x^2 > a^2$, that is, either:
:$x > a$
or:
:$x < -a$
where it is assumed that $a > 0$.
Consider the [[Definition:Arcsecant|arcsecant]] substitution:
:$u = \arcsec {\dfrac x a}$
which is defined for all $x$ such that $\size {\dfrac x a} \ge 1$.
That is:
:$\size... | Primitive of Reciprocal of x by Root of x squared minus a squared/Arcsecant Form | https://proofwiki.org/wiki/Primitive_of_Reciprocal_of_x_by_Root_of_x_squared_minus_a_squared/Arcsecant_Form | https://proofwiki.org/wiki/Primitive_of_Reciprocal_of_x_by_Root_of_x_squared_minus_a_squared/Arcsecant_Form | [
"Primitive of Reciprocal of x by Root of x squared minus a squared",
"Arcsecant Function"
] | [] | [
"Definition:Inverse Secant/Real/Arcsecant",
"Derivative of Secant Function",
"Integration by Substitution",
"Primitive of Constant Multiple of Function",
"Sum of Squares of Sine and Cosine/Corollary 1",
"Integral of Constant",
"Integration by Substitution"
] |
proofwiki-9096 | Primitive of Reciprocal of x by Root of x squared plus a squared/Logarithm Form | For $x \in \R_{\ne 0}$:
:$\ds \int \frac {\d x} {x \sqrt {x^2 + a^2} } = -\frac 1 a \map \ln {\frac a x + \frac {\sqrt {a^2 + x^2} } {\size x} } + C$ | {{begin-eqn}}
{{eqn | l = \int \frac {\d x} {x \sqrt {x^2 + a^2} }
| r = -\frac 1 a \arcsch \frac x a + C
| c = Primitive of Reciprocal of $x \sqrt {x^2 + a^2}$: $\arcsch$ form
}}
{{eqn | r = -\frac 1 a \map \ln {\frac a x + \frac {\sqrt {a^2 + x^2} } {\size x} } + C
| c = $\arcsch \dfrac x a$ in Loga... | For $x \in \R_{\ne 0}$:
:$\ds \int \frac {\d x} {x \sqrt {x^2 + a^2} } = -\frac 1 a \map \ln {\frac a x + \frac {\sqrt {a^2 + x^2} } {\size x} } + C$ | {{begin-eqn}}
{{eqn | l = \int \frac {\d x} {x \sqrt {x^2 + a^2} }
| r = -\frac 1 a \arcsch \frac x a + C
| c = [[Primitive of Reciprocal of x by Root of x squared plus a squared/Inverse Hyperbolic Cosecant Form|Primitive of Reciprocal of $x \sqrt {x^2 + a^2}$: $\arcsch$ form]]
}}
{{eqn | r = -\frac 1 a \ma... | Primitive of Reciprocal of x by Root of x squared plus a squared/Logarithm Form | https://proofwiki.org/wiki/Primitive_of_Reciprocal_of_x_by_Root_of_x_squared_plus_a_squared/Logarithm_Form | https://proofwiki.org/wiki/Primitive_of_Reciprocal_of_x_by_Root_of_x_squared_plus_a_squared/Logarithm_Form | [
"Primitive of Reciprocal of x by Root of x squared plus a squared"
] | [] | [
"Primitive of Reciprocal of x by Root of x squared plus a squared/Inverse Hyperbolic Cosecant Form",
"Real Area Hyperbolic Cosecant of x over a in Logarithm Form"
] |
proofwiki-9097 | Primitive of Reciprocal of x by Root of a squared minus x squared/Logarithm Form | For $a > 0$ and $0 < \size x < a$:
:$\ds \int \frac {\d x} {x \sqrt {a^2 - x^2} } = -\frac 1 a \map \ln {\frac {a + \sqrt {a^2 - x^2} } {\size x} } + C$ | {{begin-eqn}}
{{eqn | l = \int \frac {\d x} {x \sqrt {a^2 - x^2} }
| r = -\frac 1 a \sech^{-1} {\frac {\size x} a} + C
| c = Primitive of Reciprocal of $x \sqrt {a^2 - x^2}$: $\sech^{-1}$ form
}}
{{eqn | r = -\frac 1 a \map \ln {\frac {1 + \sqrt {1 - \paren {\frac {\size x} a}^2} } {\frac {\size x} a} } + C... | For $a > 0$ and $0 < \size x < a$:
:$\ds \int \frac {\d x} {x \sqrt {a^2 - x^2} } = -\frac 1 a \map \ln {\frac {a + \sqrt {a^2 - x^2} } {\size x} } + C$ | {{begin-eqn}}
{{eqn | l = \int \frac {\d x} {x \sqrt {a^2 - x^2} }
| r = -\frac 1 a \sech^{-1} {\frac {\size x} a} + C
| c = [[Primitive of Reciprocal of x by Root of a squared minus x squared/Inverse Hyperbolic Secant Form|Primitive of Reciprocal of $x \sqrt {a^2 - x^2}$: $\sech^{-1}$ form]]
}}
{{eqn | r =... | Primitive of Reciprocal of x by Root of a squared minus x squared/Logarithm Form | https://proofwiki.org/wiki/Primitive_of_Reciprocal_of_x_by_Root_of_a_squared_minus_x_squared/Logarithm_Form | https://proofwiki.org/wiki/Primitive_of_Reciprocal_of_x_by_Root_of_a_squared_minus_x_squared/Logarithm_Form | [
"Primitive of Reciprocal of x by Root of a squared minus x squared"
] | [] | [
"Primitive of Reciprocal of x by Root of a squared minus x squared/Inverse Hyperbolic Secant Form",
"Definition:Fraction/Numerator",
"Definition:Fraction/Denominator",
"Definition:Square Root"
] |
proofwiki-9098 | Generalized Integration by Parts | Let $\map f x, \map g x$ be real functions which are integrable and at least $n$ times differentiable.
Then:
{{begin-eqn}}
{{eqn | l = \int f^{\paren n} g \rd x
| r = \sum_{j \mathop = 0}^{n - 1} \paren {-1}^j f^{\paren {n - j - 1} } g^{\paren j} + \paren {-1}^n \int f g^{\paren n} \rd x
| c =
}}
{{eqn | r... | Proof by induction:
For all $n \in \N_{> 0}$, let $\map P n$ be the proposition:
:$\ds \int f^{\paren n} g \rd x = \sum_{j \mathop = 0}^{n - 1} \paren {-1}^j f^{\paren {n - j - 1} } g^\paren j + \paren {-1}^n \int f g^{\paren n} \rd x$ | Let $\map f x, \map g x$ be [[Definition:Real Function|real functions]] which are [[Definition:Integrable Function|integrable]] and at least [[Definition:Nth Derivative|$n$ times differentiable]].
Then:
{{begin-eqn}}
{{eqn | l = \int f^{\paren n} g \rd x
| r = \sum_{j \mathop = 0}^{n - 1} \paren {-1}^j f^{\pare... | Proof by [[Principle of Mathematical Induction|induction]]:
For all $n \in \N_{> 0}$, let $\map P n$ be the [[Definition:Proposition|proposition]]:
:$\ds \int f^{\paren n} g \rd x = \sum_{j \mathop = 0}^{n - 1} \paren {-1}^j f^{\paren {n - j - 1} } g^\paren j + \paren {-1}^n \int f g^{\paren n} \rd x$ | Generalized Integration by Parts | https://proofwiki.org/wiki/Generalized_Integration_by_Parts | https://proofwiki.org/wiki/Generalized_Integration_by_Parts | [
"Integral Calculus"
] | [
"Definition:Real Function",
"Definition:Integrable Function",
"Definition:Derivative/Higher Derivatives/Higher Order",
"Definition:Derivative/Higher Derivatives/Higher Order"
] | [
"Principle of Mathematical Induction",
"Definition:Proposition",
"Principle of Mathematical Induction"
] |
proofwiki-9099 | Primitive of Function of a x + b | :$\ds \int \map F {a x + b} \rd x = \frac 1 a \int \map F u \rd u$
where $u = a x + b$. | {{begin-eqn}}
{{eqn | l = u
| r = a x + b
| c =
}}
{{eqn | ll= \leadsto
| l = \frac {\d u} {\d x}
| r = a
| c = {{Corollary|Derivative of Function of Constant Multiple}}
}}
{{eqn | ll= \leadsto
| l = \int \map F {a x + b} \rd x
| r = \int \frac {\map F u} a \d u
| c = P... | :$\ds \int \map F {a x + b} \rd x = \frac 1 a \int \map F u \rd u$
where $u = a x + b$. | {{begin-eqn}}
{{eqn | l = u
| r = a x + b
| c =
}}
{{eqn | ll= \leadsto
| l = \frac {\d u} {\d x}
| r = a
| c = {{Corollary|Derivative of Function of Constant Multiple}}
}}
{{eqn | ll= \leadsto
| l = \int \map F {a x + b} \rd x
| r = \int \frac {\map F u} a \d u
| c = [... | Primitive of Function of a x + b | https://proofwiki.org/wiki/Primitive_of_Function_of_a_x_+_b | https://proofwiki.org/wiki/Primitive_of_Function_of_a_x_+_b | [
"Integral Substitutions",
"Primitives involving a x + b"
] | [] | [
"Primitive of Composite Function",
"Primitive of Constant Multiple of Function"
] |
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