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-9700 | Primitive of Power of x by Arccosine of x over a | :$\ds \int x^m \arccos \frac x a \rd x = \frac {x^{m + 1} } {m + 1} \arccos \frac x a + \frac 1 {m + 1} \int \frac {x^{m + 1} \rd x} {\sqrt {a^2 - x^2} }$ | 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 = \arccos \frac x a
| c =
}}
{{eqn | ll= \leadsto
| l = \frac {\d u} {\d x}
| r = \frac {-1} {\sqrt {a^2 - x^2} }
| c ... | :$\ds \int x^m \arccos \frac x a \rd x = \frac {x^{m + 1} } {m + 1} \arccos \frac x a + \frac 1 {m + 1} \int \frac {x^{m + 1} \rd x} {\sqrt {a^2 - x^2} }$ | 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 = \arccos \frac x a
| c =
}}
{{eqn | ll= \leadsto
| l = \frac {\d u} {\d x}
| r = \frac {-1} {\sqr... | Primitive of Power of x by Arccosine of x over a/Proof 2 | https://proofwiki.org/wiki/Primitive_of_Power_of_x_by_Arccosine_of_x_over_a | https://proofwiki.org/wiki/Primitive_of_Power_of_x_by_Arccosine_of_x_over_a/Proof_2 | [
"Primitive of Power of x by Arccosine of x over a",
"Primitives involving Inverse Cosine Function"
] | [] | [
"Definition:Primitive",
"Derivative of Arccosine Function/Corollary",
"Primitive of Power",
"Integration by Parts",
"Primitive of Constant Multiple of Function"
] |
proofwiki-9701 | Primitive of Power of x by Arctangent of x over a | :$\ds \int x^m \arctan \frac x a \rd x = \frac {x^{m + 1} } {m + 1} \arctan \frac x a - \frac a {m + 1} \int \frac {x^{m + 1} \rd x} {x^2 + a^2}$ | 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... | :$\ds \int x^m \arctan \frac x a \rd x = \frac {x^{m + 1} } {m + 1} \arctan \frac x a - \frac a {m + 1} \int \frac {x^{m + 1} \rd x} {x^2 + a^2}$ | 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_of_x_by_Arctangent_of_x_over_a | https://proofwiki.org/wiki/Primitive_of_Power_of_x_by_Arctangent_of_x_over_a/Proof_1 | [
"Primitive of Power of x by Arctangent of x over a",
"Primitives involving Inverse Tangent Function"
] | [] | [
"Primitive of Power of x by Arctangent of x",
"Primitive of Constant Multiple of Function",
"Primitive of Function of Constant Multiple"
] |
proofwiki-9702 | Primitive of Power of x by Arctangent of x over a | :$\ds \int x^m \arctan \frac x a \rd x = \frac {x^{m + 1} } {m + 1} \arctan \frac x a - \frac a {m + 1} \int \frac {x^{m + 1} \rd x} {x^2 + a^2}$ | 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... | :$\ds \int x^m \arctan \frac x a \rd x = \frac {x^{m + 1} } {m + 1} \arctan \frac x a - \frac a {m + 1} \int \frac {x^{m + 1} \rd x} {x^2 + a^2}$ | 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_of_x_by_Arctangent_of_x_over_a | https://proofwiki.org/wiki/Primitive_of_Power_of_x_by_Arctangent_of_x_over_a/Proof_2 | [
"Primitive of Power of x by Arctangent of x over a",
"Primitives involving Inverse Tangent Function"
] | [] | [
"Definition:Primitive",
"Derivative of Arctangent Function/Corollary",
"Primitive of Power",
"Integration by Parts",
"Primitive of Constant Multiple of Function"
] |
proofwiki-9703 | Primitive of Power of x by Arccotangent of x over a | :$\ds \int x^m \arccot \frac x a \rd x = \frac {x^{m + 1} } {m + 1} \arccot \frac x a + \frac a {m + 1} \int \frac {x^{m + 1} \rd x} {x^2 + a^2}$ | Recall:
{{begin-eqn}}
{{eqn | n = 1
| l = \int x^m \arccot x \rd x
| r = \frac {x^{m + 1} } {m + 1} \arccot x + \frac 1 {m + 1} \int \frac {x^{m + 1} \rd x} {x^2 + 1}
| c = Primitive of $x^m \arccot x$
}}
{{end-eqn}}
Then:
{{begin-eqn}}
{{eqn | l = \int x^m \arccot \frac x a \rd x
| r = \int a^m... | :$\ds \int x^m \arccot \frac x a \rd x = \frac {x^{m + 1} } {m + 1} \arccot \frac x a + \frac a {m + 1} \int \frac {x^{m + 1} \rd x} {x^2 + a^2}$ | Recall:
{{begin-eqn}}
{{eqn | n = 1
| l = \int x^m \arccot x \rd x
| r = \frac {x^{m + 1} } {m + 1} \arccot x + \frac 1 {m + 1} \int \frac {x^{m + 1} \rd x} {x^2 + 1}
| c = [[Primitive of Power of x by Arccotangent of x|Primitive of $x^m \arccot x$]]
}}
{{end-eqn}}
Then:
{{begin-eqn}}
{{eqn | l = \... | Primitive of Power of x by Arccotangent of x over a/Proof 1 | https://proofwiki.org/wiki/Primitive_of_Power_of_x_by_Arccotangent_of_x_over_a | https://proofwiki.org/wiki/Primitive_of_Power_of_x_by_Arccotangent_of_x_over_a/Proof_1 | [
"Primitive of Power of x by Arccotangent of x over a",
"Primitives involving Inverse Cotangent Function"
] | [] | [
"Primitive of Power of x by Arccotangent of x",
"Primitive of Constant Multiple of Function",
"Primitive of Function of Constant Multiple"
] |
proofwiki-9704 | Primitive of Power of x by Arccotangent of x over a | :$\ds \int x^m \arccot \frac x a \rd x = \frac {x^{m + 1} } {m + 1} \arccot \frac x a + \frac a {m + 1} \int \frac {x^{m + 1} \rd x} {x^2 + a^2}$ | 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 = \arccot \frac x a
| c =
}}
{{eqn | ll= \leadsto
| l = \frac {\d u} {\d x}
| r = \frac {-a} {x^2 + a^2}
| c = Derivat... | :$\ds \int x^m \arccot \frac x a \rd x = \frac {x^{m + 1} } {m + 1} \arccot \frac x a + \frac a {m + 1} \int \frac {x^{m + 1} \rd x} {x^2 + a^2}$ | 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 = \arccot \frac x a
| c =
}}
{{eqn | ll= \leadsto
| l = \frac {\d u} {\d x}
| r = \frac {-a} {x^2 ... | Primitive of Power of x by Arccotangent of x over a/Proof 2 | https://proofwiki.org/wiki/Primitive_of_Power_of_x_by_Arccotangent_of_x_over_a | https://proofwiki.org/wiki/Primitive_of_Power_of_x_by_Arccotangent_of_x_over_a/Proof_2 | [
"Primitive of Power of x by Arccotangent of x over a",
"Primitives involving Inverse Cotangent Function"
] | [] | [
"Definition:Primitive",
"Derivative of Arccotangent Function/Corollary",
"Primitive of Power",
"Integration by Parts",
"Primitive of Constant Multiple of Function"
] |
proofwiki-9705 | Primitive of Power of x by Arcsecant of x over a | :<nowiki> $\ds \int x^m \arcsec \frac x a \rd x = \begin {cases}
\dfrac {x^{m + 1} } {m + 1} \arcsec \dfrac x a - \dfrac a {m + 1} \ds \int \dfrac {x^m \rd x} {\sqrt {x^2 - a^2} } & : 0 < \arcsec \dfrac x a < \dfrac \pi 2 \\
\dfrac {x^{m + 1} } {m + 1} \arcsec \dfrac x a + \dfrac a {m + 1} \ds \int \dfrac {x^m \rd x} {... | Recall:
{{begin-eqn}}
{{eqn | n = 1
| l = \int x^m \arcsec x \rd x
| r = <nowiki>\begin {cases}
\dfrac {x^{m + 1} } {m + 1} \arcsec x - \dfrac 1 {m + 1} \ds \int \dfrac {x^m \rd x} {\sqrt {x^2 - 1} } & : 0 < \arcsec x < \dfrac \pi 2 \\
\dfrac {x^{m + 1} } {m + 1} \arcsec x + \dfrac 1 {m + 1} \ds \int \dfrac... | :<nowiki> $\ds \int x^m \arcsec \frac x a \rd x = \begin {cases}
\dfrac {x^{m + 1} } {m + 1} \arcsec \dfrac x a - \dfrac a {m + 1} \ds \int \dfrac {x^m \rd x} {\sqrt {x^2 - a^2} } & : 0 < \arcsec \dfrac x a < \dfrac \pi 2 \\
\dfrac {x^{m + 1} } {m + 1} \arcsec \dfrac x a + \dfrac a {m + 1} \ds \int \dfrac {x^m \rd x} {... | Recall:
{{begin-eqn}}
{{eqn | n = 1
| l = \int x^m \arcsec x \rd x
| r = <nowiki>\begin {cases}
\dfrac {x^{m + 1} } {m + 1} \arcsec x - \dfrac 1 {m + 1} \ds \int \dfrac {x^m \rd x} {\sqrt {x^2 - 1} } & : 0 < \arcsec x < \dfrac \pi 2 \\
\dfrac {x^{m + 1} } {m + 1} \arcsec x + \dfrac 1 {m + 1} \ds \int \dfra... | Primitive of Power of x by Arcsecant of x over a/Proof 1 | https://proofwiki.org/wiki/Primitive_of_Power_of_x_by_Arcsecant_of_x_over_a | https://proofwiki.org/wiki/Primitive_of_Power_of_x_by_Arcsecant_of_x_over_a/Proof_1 | [
"Primitive of Power of x by Arcsecant of x over a",
"Primitives involving Inverse Secant Function"
] | [] | [
"Primitive of Power of x by Arcsecant of x",
"Primitive of Constant Multiple of Function",
"Primitive of Function of Constant Multiple"
] |
proofwiki-9706 | Primitive of Power of x by Arcsecant of x over a | :<nowiki> $\ds \int x^m \arcsec \frac x a \rd x = \begin {cases}
\dfrac {x^{m + 1} } {m + 1} \arcsec \dfrac x a - \dfrac a {m + 1} \ds \int \dfrac {x^m \rd x} {\sqrt {x^2 - a^2} } & : 0 < \arcsec \dfrac x a < \dfrac \pi 2 \\
\dfrac {x^{m + 1} } {m + 1} \arcsec \dfrac x a + \dfrac a {m + 1} \ds \int \dfrac {x^m \rd x} {... | With a view to expressing the primitive in the form:
:$\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 = \arcsec \frac x a
| c =
}}
{{eqn | ll= \leadsto
| l = \frac {\d u} {\d x}
| r = <nowiki> \begin {cases} \dfrac a {x \sqrt ... | :<nowiki> $\ds \int x^m \arcsec \frac x a \rd x = \begin {cases}
\dfrac {x^{m + 1} } {m + 1} \arcsec \dfrac x a - \dfrac a {m + 1} \ds \int \dfrac {x^m \rd x} {\sqrt {x^2 - a^2} } & : 0 < \arcsec \dfrac x a < \dfrac \pi 2 \\
\dfrac {x^{m + 1} } {m + 1} \arcsec \dfrac x a + \dfrac a {m + 1} \ds \int \dfrac {x^m \rd x} {... | With a view to expressing the [[Definition:Primitive|primitive]] in the form:
:$\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 = \arcsec \frac x a
| c =
}}
{{eqn | ll= \leadsto
| l = \frac {\d u} {\d x}
| r = <nowiki> \begin ... | Primitive of Power of x by Arcsecant of x over a/Proof 2 | https://proofwiki.org/wiki/Primitive_of_Power_of_x_by_Arcsecant_of_x_over_a | https://proofwiki.org/wiki/Primitive_of_Power_of_x_by_Arcsecant_of_x_over_a/Proof_2 | [
"Primitive of Power of x by Arcsecant of x over a",
"Primitives involving Inverse Secant Function"
] | [] | [
"Definition:Primitive",
"Derivative of Arcsecant Function/Corollary 1",
"Primitive of Power",
"Definition:Real Interval/Open",
"Integration by Parts",
"Primitive of Constant Multiple of Function",
"Definition:Real Interval/Open",
"Integration by Parts",
"Primitive of Constant Multiple of Function"
] |
proofwiki-9707 | Primitive of Power of x by Arccosecant of x over a | :<nowiki>$\ds \int x^m \arccsc \frac x a \rd x = \begin {cases}
\ds \dfrac {x^{m + 1} } {m + 1} \arccsc \dfrac x a + \dfrac a {m + 1} \int \dfrac {x^m \rd x} {\sqrt {x^2 - a^2} } & : 0 < \arccsc \dfrac x a < \dfrac \pi 2 \\ \\
\ds \dfrac {x^{m + 1} } {m + 1} \arccsc \dfrac x a - \dfrac a {m + 1} \int \dfrac {x^m \rd x}... | Recall:
{{begin-eqn}}
{{eqn | n = 1
| l = \int x^m \arccsc x \rd x
| r = <nowiki>\begin {cases}
\ds \dfrac {x^{m + 1} } {m + 1} \arccsc x + \dfrac 1 {m + 1} \int \dfrac {x^m \rd x} {\sqrt {x^2 - 1} } & : 0 < \arccsc x < \dfrac \pi 2 \\ \\
\ds \dfrac {x^{m + 1} } {m + 1} \arccsc x - \dfrac 1 {m + 1} \int \df... | :<nowiki>$\ds \int x^m \arccsc \frac x a \rd x = \begin {cases}
\ds \dfrac {x^{m + 1} } {m + 1} \arccsc \dfrac x a + \dfrac a {m + 1} \int \dfrac {x^m \rd x} {\sqrt {x^2 - a^2} } & : 0 < \arccsc \dfrac x a < \dfrac \pi 2 \\ \\
\ds \dfrac {x^{m + 1} } {m + 1} \arccsc \dfrac x a - \dfrac a {m + 1} \int \dfrac {x^m \rd x}... | Recall:
{{begin-eqn}}
{{eqn | n = 1
| l = \int x^m \arccsc x \rd x
| r = <nowiki>\begin {cases}
\ds \dfrac {x^{m + 1} } {m + 1} \arccsc x + \dfrac 1 {m + 1} \int \dfrac {x^m \rd x} {\sqrt {x^2 - 1} } & : 0 < \arccsc x < \dfrac \pi 2 \\ \\
\ds \dfrac {x^{m + 1} } {m + 1} \arccsc x - \dfrac 1 {m + 1} \int \d... | Primitive of Power of x by Arccosecant of x over a/Proof 1 | https://proofwiki.org/wiki/Primitive_of_Power_of_x_by_Arccosecant_of_x_over_a | https://proofwiki.org/wiki/Primitive_of_Power_of_x_by_Arccosecant_of_x_over_a/Proof_1 | [
"Primitive of Power of x by Arccosecant of x over a",
"Primitives involving Inverse Cosecant Function"
] | [] | [
"Primitive of Power of x by Arccosecant of x",
"Primitive of Constant Multiple of Function",
"Primitive of Function of Constant Multiple"
] |
proofwiki-9708 | Primitive of Power of x by Arccosecant of x over a | :<nowiki>$\ds \int x^m \arccsc \frac x a \rd x = \begin {cases}
\ds \dfrac {x^{m + 1} } {m + 1} \arccsc \dfrac x a + \dfrac a {m + 1} \int \dfrac {x^m \rd x} {\sqrt {x^2 - a^2} } & : 0 < \arccsc \dfrac x a < \dfrac \pi 2 \\ \\
\ds \dfrac {x^{m + 1} } {m + 1} \arccsc \dfrac x a - \dfrac a {m + 1} \int \dfrac {x^m \rd x}... | With a view to expressing the primitive in the form:
:$\ds \int u \frac {\rd v} {\d x} \rd x = u v - \int v \frac {\d u} {\d x} \rd x$
let:
{{begin-eqn}}
{{eqn | l = u
| r = \arccsc \frac x a
| c =
}}
{{eqn | ll= \leadsto
| l = \frac {\d u} {\d x}
| r = <nowiki> \begin {cases} \dfrac {-a} {x \s... | :<nowiki>$\ds \int x^m \arccsc \frac x a \rd x = \begin {cases}
\ds \dfrac {x^{m + 1} } {m + 1} \arccsc \dfrac x a + \dfrac a {m + 1} \int \dfrac {x^m \rd x} {\sqrt {x^2 - a^2} } & : 0 < \arccsc \dfrac x a < \dfrac \pi 2 \\ \\
\ds \dfrac {x^{m + 1} } {m + 1} \arccsc \dfrac x a - \dfrac a {m + 1} \int \dfrac {x^m \rd x}... | With a view to expressing the [[Definition:Primitive|primitive]] in the form:
:$\ds \int u \frac {\rd v} {\d x} \rd x = u v - \int v \frac {\d u} {\d x} \rd x$
let:
{{begin-eqn}}
{{eqn | l = u
| r = \arccsc \frac x a
| c =
}}
{{eqn | ll= \leadsto
| l = \frac {\d u} {\d x}
| r = <nowiki> \begin... | Primitive of Power of x by Arccosecant of x over a/Proof 2 | https://proofwiki.org/wiki/Primitive_of_Power_of_x_by_Arccosecant_of_x_over_a | https://proofwiki.org/wiki/Primitive_of_Power_of_x_by_Arccosecant_of_x_over_a/Proof_2 | [
"Primitive of Power of x by Arccosecant of x over a",
"Primitives involving Inverse Cosecant Function"
] | [] | [
"Definition:Primitive",
"Derivative of Arccosecant Function/Corollary",
"Primitive of Power",
"Definition:Real Interval/Open",
"Integration by Parts",
"Primitive of Constant Multiple of Function",
"Definition:Real Interval/Open",
"Integration by Parts",
"Primitive of Constant Multiple of Function"
] |
proofwiki-9709 | Primitive of Exponential of a x | :$\ds \int e^{a x} \rd x = \frac {e^{a x} } a + C$ | {{begin-eqn}}
{{eqn | l = \int e^{a x} \cos b x \rd x
| r = \frac {e^{a x} \cos b x} a + \frac b a \int e^{a x} \sin b x \rd x
| c = Primitive of $e^{a x} \cos b x$: Lemma
}}
{{eqn | r = \frac {e^{a x} \cos b x} a + \frac b a \paren {\frac {e^{a x} \sin b x} a - \frac b a \int e^{a x} \cos b x \rd x}
... | :$\ds \int e^{a x} \rd x = \frac {e^{a x} } a + C$ | {{begin-eqn}}
{{eqn | l = \int e^{a x} \cos b x \rd x
| r = \frac {e^{a x} \cos b x} a + \frac b a \int e^{a x} \sin b x \rd x
| c = [[Primitive of Exponential of a x by Cosine of b x/Lemma|Primitive of $e^{a x} \cos b x$: Lemma]]
}}
{{eqn | r = \frac {e^{a x} \cos b x} a + \frac b a \paren {\frac {e^{a x} ... | Primitive of Exponential of a x by Cosine of b x/Proof 1 | https://proofwiki.org/wiki/Primitive_of_Exponential_of_a_x | https://proofwiki.org/wiki/Primitive_of_Exponential_of_a_x_by_Cosine_of_b_x/Proof_1 | [
"Primitives involving Exponential Function",
"Primitive of Exponential of a x"
] | [] | [
"Primitive of Exponential of a x by Cosine of b x/Lemma",
"Primitive of Exponential of a x by Sine of b x/Lemma"
] |
proofwiki-9710 | Primitive of Exponential of a x | :$\ds \int e^{a x} \rd x = \frac {e^{a x} } a + C$ | {{begin-eqn}}
{{eqn | l = \int e^{a x} e^{i b x} \rd x
| r = i \int e^{a x} \sin b x \rd x + \int e^{a x} \cos b x \rd x
| c = Euler's Formula
}}
{{eqn | ll= \leadsto
| l = \int e^{a x} \cos b x \rd x
| r = \map \Re {\int e^{\paren {a + i b} x} \rd x}
}}
{{eqn | r = \map \Re {\frac {e^{\paren {... | :$\ds \int e^{a x} \rd x = \frac {e^{a x} } a + C$ | {{begin-eqn}}
{{eqn | l = \int e^{a x} e^{i b x} \rd x
| r = i \int e^{a x} \sin b x \rd x + \int e^{a x} \cos b x \rd x
| c = [[Euler's Formula]]
}}
{{eqn | ll= \leadsto
| l = \int e^{a x} \cos b x \rd x
| r = \map \Re {\int e^{\paren {a + i b} x} \rd x}
}}
{{eqn | r = \map \Re {\frac {e^{\par... | Primitive of Exponential of a x by Cosine of b x/Proof 2 | https://proofwiki.org/wiki/Primitive_of_Exponential_of_a_x | https://proofwiki.org/wiki/Primitive_of_Exponential_of_a_x_by_Cosine_of_b_x/Proof_2 | [
"Primitives involving Exponential Function",
"Primitive of Exponential of a x"
] | [] | [
"Euler's Formula",
"Primitive of Exponential of a x",
"Euler's Formula",
"Definition:Complex Number/Real Part"
] |
proofwiki-9711 | Primitive of Exponential of a x | :$\ds \int e^{a x} \rd x = \frac {e^{a x} } a + C$ | Let $a, b \in \R_{>0}$ be real constants.
Let $f_1$ and $f_2$ be the real functions defined as:
{{begin-eqn}}
{{eqn | q = \forall x \in \R
| l = \map {f_1} x
| r = \map \exp {a x} \map \cos {b x}
}}
{{eqn | l = \map {f_2} x
| r = \map \exp {a x} \map \sin {b x}
}}
{{end-eqn}}
Let $\map \CC \R$ denote ... | :$\ds \int e^{a x} \rd x = \frac {e^{a x} } a + C$ | Let $a, b \in \R_{>0}$ be [[Definition:Real Number|real]] [[Definition:Constant|constants]].
Let $f_1$ and $f_2$ be the [[Definition:Real Function|real functions]] defined as:
{{begin-eqn}}
{{eqn | q = \forall x \in \R
| l = \map {f_1} x
| r = \map \exp {a x} \map \cos {b x}
}}
{{eqn | l = \map {f_2} x
... | Primitive of Exponential of a x by Cosine of b x/Proof 3 | https://proofwiki.org/wiki/Primitive_of_Exponential_of_a_x | https://proofwiki.org/wiki/Primitive_of_Exponential_of_a_x_by_Cosine_of_b_x/Proof_3 | [
"Primitives involving Exponential Function",
"Primitive of Exponential of a x"
] | [] | [
"Definition:Real Number",
"Definition:Constant",
"Definition:Real Function",
"Definition:Continuous Real-Valued Function Space",
"Space of Continuously Differentiable on Closed Interval Real-Valued Functions with Pointwise Addition and Pointwise Scalar Multiplication forms Vector Space",
"Definition:Vecto... |
proofwiki-9712 | Primitive of Exponential of a x | :$\ds \int e^{a x} \rd x = \frac {e^{a x} } a + C$ | {{begin-eqn}}
{{eqn | l = \int e^{a x} \sin b x \rd x
| r = \frac {e^{a x} \sin b x} a - \frac b a \int e^{a x} \cos b x \rd x
| c = Primitive of $e^{a x} \sin b x$: Lemma
}}
{{eqn | r = \frac {e^{a x} \sin b x} a - \frac b a \paren {\frac {e^{a x} \cos b x} a + \frac b a \int e^{a x} \sin b x \rd x}
... | :$\ds \int e^{a x} \rd x = \frac {e^{a x} } a + C$ | {{begin-eqn}}
{{eqn | l = \int e^{a x} \sin b x \rd x
| r = \frac {e^{a x} \sin b x} a - \frac b a \int e^{a x} \cos b x \rd x
| c = [[Primitive of Exponential of a x by Sine of b x/Lemma|Primitive of $e^{a x} \sin b x$: Lemma]]
}}
{{eqn | r = \frac {e^{a x} \sin b x} a - \frac b a \paren {\frac {e^{a x} \c... | Primitive of Exponential of a x by Sine of b x/Proof 1 | https://proofwiki.org/wiki/Primitive_of_Exponential_of_a_x | https://proofwiki.org/wiki/Primitive_of_Exponential_of_a_x_by_Sine_of_b_x/Proof_1 | [
"Primitives involving Exponential Function",
"Primitive of Exponential of a x"
] | [] | [
"Primitive of Exponential of a x by Sine of b x/Lemma",
"Primitive of Exponential of a x by Cosine of b x/Lemma",
"Definition:Common Denominator"
] |
proofwiki-9713 | Primitive of Exponential of a x | :$\ds \int e^{a x} \rd x = \frac {e^{a x} } a + C$ | {{begin-eqn}}
{{eqn | l = \cos b x + i \sin b x
| r = e^{i b x}
| c = Euler's Formula
}}
{{eqn | ll= \leadsto
| l = e^{a x} \cos b x + i e^{a x} \sin b x
| r = e^{a x} e^{i b x}
| c = multiplying both sides by $e^{a x}$
}}
{{eqn | r = e^{\paren {a + i b} x}
| c = Exponent Combination... | :$\ds \int e^{a x} \rd x = \frac {e^{a x} } a + C$ | {{begin-eqn}}
{{eqn | l = \cos b x + i \sin b x
| r = e^{i b x}
| c = [[Euler's Formula]]
}}
{{eqn | ll= \leadsto
| l = e^{a x} \cos b x + i e^{a x} \sin b x
| r = e^{a x} e^{i b x}
| c = multiplying both sides by $e^{a x}$
}}
{{eqn | r = e^{\paren {a + i b} x}
| c = [[Exponent Combi... | Primitive of Exponential of a x by Sine of b x/Proof 2 | https://proofwiki.org/wiki/Primitive_of_Exponential_of_a_x | https://proofwiki.org/wiki/Primitive_of_Exponential_of_a_x_by_Sine_of_b_x/Proof_2 | [
"Primitives involving Exponential Function",
"Primitive of Exponential of a x"
] | [] | [
"Euler's Formula",
"Exponent Combination Laws",
"Linear Combination of Complex Integrals",
"Primitive of Exponential of a x",
"Exponent Combination Laws",
"Euler's Formula"
] |
proofwiki-9714 | Primitive of Exponential of a x | :$\ds \int e^{a x} \rd x = \frac {e^{a x} } a + C$ | {{begin-eqn}}
{{eqn | l = \int e^{a x} \sin b x \rd x
| r = \int e^{a x} \paren {\frac {e^{i b x} - e^{-i b x} } {2 i} } \rd x
| c = Euler's Sine Identity
}}
{{eqn | r = \frac 1 {2 i} \int e^{a x} \paren {e^{i b x} - e^{-i b x} } \rd x
| c = Primitive of Constant Multiple of Function
}}
{{eqn | r = \f... | :$\ds \int e^{a x} \rd x = \frac {e^{a x} } a + C$ | {{begin-eqn}}
{{eqn | l = \int e^{a x} \sin b x \rd x
| r = \int e^{a x} \paren {\frac {e^{i b x} - e^{-i b x} } {2 i} } \rd x
| c = [[Euler's Sine Identity]]
}}
{{eqn | r = \frac 1 {2 i} \int e^{a x} \paren {e^{i b x} - e^{-i b x} } \rd x
| c = [[Primitive of Constant Multiple of Function]]
}}
{{eqn ... | Primitive of Exponential of a x by Sine of b x/Proof 3 | https://proofwiki.org/wiki/Primitive_of_Exponential_of_a_x | https://proofwiki.org/wiki/Primitive_of_Exponential_of_a_x_by_Sine_of_b_x/Proof_3 | [
"Primitives involving Exponential Function",
"Primitive of Exponential of a x"
] | [] | [
"Euler's Sine Identity",
"Primitive of Constant Multiple of Function",
"Exponent Combination Laws/Product of Powers",
"Linear Combination of Integrals/Indefinite",
"Primitive of Exponential of a x",
"Exponent Combination Laws/Product of Powers",
"Product of Complex Number with Conjugate",
"Euler's Cos... |
proofwiki-9715 | Primitive of Exponential of a x | :$\ds \int e^{a x} \rd x = \frac {e^{a x} } a + C$ | Let $a, b, x \in \R$ be real numbers.
Suppose $a \ne 0 \ne b$.
Let $f_1$ and $f_2$ be the real functions defined as:
{{begin-eqn}}
{{eqn | q = \forall x \in \R
| l = \map {f_1} x
| r = \map \exp {a x} \map \cos {b x}
}}
{{eqn | l = \map {f_2} x
| r = \map \exp {a x} \map \sin {b x}
}}
{{end-eqn}}
Let ... | :$\ds \int e^{a x} \rd x = \frac {e^{a x} } a + C$ | Let $a, b, x \in \R$ be [[Definition:Real Number|real numbers]].
Suppose $a \ne 0 \ne b$.
Let $f_1$ and $f_2$ be the [[Definition:Real Function|real functions]] defined as:
{{begin-eqn}}
{{eqn | q = \forall x \in \R
| l = \map {f_1} x
| r = \map \exp {a x} \map \cos {b x}
}}
{{eqn | l = \map {f_2} x
... | Primitive of Exponential of a x by Sine of b x/Proof 4 | https://proofwiki.org/wiki/Primitive_of_Exponential_of_a_x | https://proofwiki.org/wiki/Primitive_of_Exponential_of_a_x_by_Sine_of_b_x/Proof_4 | [
"Primitives involving Exponential Function",
"Primitive of Exponential of a x"
] | [] | [
"Definition:Real Number",
"Definition:Real Function",
"Definition:Continuous Real-Valued Function Space",
"Space of Continuously Differentiable on Closed Interval Real-Valued Functions with Pointwise Addition and Pointwise Scalar Multiplication forms Vector Space",
"Definition:Vector Space",
"Definition:D... |
proofwiki-9716 | Primitive of x by Exponential of a x | :$\ds \int x e^{a x} \rd x = \frac {e^{a x} } a \paren {x - \frac 1 a} + C$ | 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 = x
| c =
}}
{{eqn | ll= \leadsto
| l = \frac {\d u} {\d x}
| r = 1
| c = Derivative of Identity Function
}}
{{end-eqn... | :$\ds \int x e^{a x} \rd x = \frac {e^{a x} } a \paren {x - \frac 1 a} + C$ | 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 = x
| c =
}}
{{eqn | ll= \leadsto
| l = \frac {\d u} {\d x}
| r = 1
| c = [[Derivative of Id... | Primitive of x by Exponential of a x | https://proofwiki.org/wiki/Primitive_of_x_by_Exponential_of_a_x | https://proofwiki.org/wiki/Primitive_of_x_by_Exponential_of_a_x | [
"Primitive of x by Exponential of a x",
"Primitives involving Exponential Function"
] | [] | [
"Definition:Primitive",
"Derivative of Identity Function",
"Primitive of Exponential of a x",
"Integration by Parts",
"Primitive of Constant Multiple of Function",
"Primitive of Exponential of a x"
] |
proofwiki-9717 | Primitive of x squared by Exponential of a x | :$\ds \int x^2 e^{a x} \rd x = \frac {e^{a x} } a \paren {x^2 - \frac {2 x} a + \frac 2 {a^2} } + C$ | 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 = x^2
| c =
}}
{{eqn | ll= \leadsto
| l = \frac {\d u} {\d x}
| r = 2 x
| c = Derivative of Power
}}
{{end-eqn}}
and l... | :$\ds \int x^2 e^{a x} \rd x = \frac {e^{a x} } a \paren {x^2 - \frac {2 x} a + \frac 2 {a^2} } + C$ | 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 = x^2
| c =
}}
{{eqn | ll= \leadsto
| l = \frac {\d u} {\d x}
| r = 2 x
| c = [[Derivative o... | Primitive of x squared by Exponential of a x | https://proofwiki.org/wiki/Primitive_of_x_squared_by_Exponential_of_a_x | https://proofwiki.org/wiki/Primitive_of_x_squared_by_Exponential_of_a_x | [
"Primitive of x squared by Exponential of a x",
"Primitives involving Exponential Function"
] | [] | [
"Definition:Primitive",
"Power Rule for Derivatives",
"Primitive of Exponential of a x",
"Integration by Parts",
"Primitive of Constant Multiple of Function",
"Primitive of x by Exponential of a x"
] |
proofwiki-9718 | Primitive of Power of x by Exponential of a x | Let $n$ be a positive integer.
Let $a$ be a non-zero real number.
Then:
{{begin-eqn}}
{{eqn | l = \int x^n e^{a x} \rd x
| r = \frac {e^{a x} } a \paren {x^n - \dfrac {n x^{n - 1} } a + \dfrac {n \paren {n - 1} x^{n - 2} } {a^2} - \dfrac {n \paren {n - 1} \paren {n - 2} x^{n - 3} } {a^3} + \cdots + \dfrac {\paren... | Proof by induction:
For all $n \in \N_{> 0}$, let $\map P n$ be the proposition:
:$\ds \int x^n e^{a x} \rd x = \frac {e^{a x} } a \sum_{k \mathop = 0}^n \paren {\paren {-1}^k \frac {n^{\underline k} x^{n - k} } {a^k} } + C$
$\map P 0$ is true, as from Primitive of $e^{a x}$:
:$\ds \int e^{a x} \rd x = \frac {e^{a x} }... | Let $n$ be a [[Definition:Positive Integer|positive integer]].
Let $a$ be a non-zero [[Definition:Real Number|real number]].
Then:
{{begin-eqn}}
{{eqn | l = \int x^n e^{a x} \rd x
| r = \frac {e^{a x} } a \paren {x^n - \dfrac {n x^{n - 1} } a + \dfrac {n \paren {n - 1} x^{n - 2} } {a^2} - \dfrac {n \paren {n - ... | Proof by [[Principle of Mathematical Induction|induction]]:
For all $n \in \N_{> 0}$, let $\map P n$ be the [[Definition:Proposition|proposition]]:
:$\ds \int x^n e^{a x} \rd x = \frac {e^{a x} } a \sum_{k \mathop = 0}^n \paren {\paren {-1}^k \frac {n^{\underline k} x^{n - k} } {a^k} } + C$
$\map P 0$ is true, as fr... | Primitive of Power of x by Exponential of a x | https://proofwiki.org/wiki/Primitive_of_Power_of_x_by_Exponential_of_a_x | https://proofwiki.org/wiki/Primitive_of_Power_of_x_by_Exponential_of_a_x | [
"Primitive of Power of x by Exponential of a x",
"Primitives involving Exponential Function"
] | [
"Definition:Positive/Integer",
"Definition:Real Number",
"Definition:Falling Factorial"
] | [
"Principle of Mathematical Induction",
"Definition:Proposition",
"Primitive of Exponential of a x",
"Principle of Mathematical Induction"
] |
proofwiki-9719 | Primitive of Exponential of a x over x | For $x > 0$:
{{begin-eqn}}
{{eqn | l = \int \frac {e^{a x} \rd x} x
| r = \ln x + \sum_{k \mathop \ge 1} \frac {\paren {a x}^k} {k \times k!} + C
| c =
}}
{{eqn | r = \ln x + \dfrac {a x} {1 \times 1!} + \dfrac {\paren {a x}^2} {2 \times 2!} + \dfrac {\paren {a x}^3} {3 \times 3!} + \cdots + C
| c = ... | {{begin-eqn}}
{{eqn | l = \int \frac {e^{a x} \rd x} x
| r = \int \frac 1 x \paren {\sum_{k \mathop = 0}^\infty \frac {\paren {a x}^k} {k!} } \rd x
| c = Power Series Expansion for Exponential Function
}}
{{eqn | r = \int \frac 1 x \paren {1 + \sum_{k \mathop = 1}^\infty \frac {\paren {a x}^k} {k!} } \rd x
... | For $x > 0$:
{{begin-eqn}}
{{eqn | l = \int \frac {e^{a x} \rd x} x
| r = \ln x + \sum_{k \mathop \ge 1} \frac {\paren {a x}^k} {k \times k!} + C
| c =
}}
{{eqn | r = \ln x + \dfrac {a x} {1 \times 1!} + \dfrac {\paren {a x}^2} {2 \times 2!} + \dfrac {\paren {a x}^3} {3 \times 3!} + \cdots + C
| c =... | {{begin-eqn}}
{{eqn | l = \int \frac {e^{a x} \rd x} x
| r = \int \frac 1 x \paren {\sum_{k \mathop = 0}^\infty \frac {\paren {a x}^k} {k!} } \rd x
| c = [[Power Series Expansion for Exponential Function]]
}}
{{eqn | r = \int \frac 1 x \paren {1 + \sum_{k \mathop = 1}^\infty \frac {\paren {a x}^k} {k!} } \r... | Primitive of Exponential of a x over x | https://proofwiki.org/wiki/Primitive_of_Exponential_of_a_x_over_x | https://proofwiki.org/wiki/Primitive_of_Exponential_of_a_x_over_x | [
"Primitives involving Exponential Function"
] | [] | [
"Power Series Expansion for Exponential Function",
"Linear Combination of Integrals/Indefinite",
"Primitive of Reciprocal",
"Definition:Negative/Real Number",
"Primitive of Power",
"Definition:Absolutely Convergent Series"
] |
proofwiki-9720 | Primitive of Exponential of a x over Power of x | :$\ds \int \frac {e^{a x} \rd x} {x^n} = \frac {-e^{a x} } {\paren {n - 1} x^{n - 1} } + \frac a {n - 1} \int \frac {e^{a x} \rd x} {x^{n - 1} } + C$
where $n \ne 1$. | 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 = e^{a x}
| c =
}}
{{eqn | ll= \leadsto
| l = \frac {\d u} {\d x}
| r = a e^{a x}
| c = Derivative of $e^{a x}$
}}
{{e... | :$\ds \int \frac {e^{a x} \rd x} {x^n} = \frac {-e^{a x} } {\paren {n - 1} x^{n - 1} } + \frac a {n - 1} \int \frac {e^{a x} \rd x} {x^{n - 1} } + C$
where $n \ne 1$. | 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 = e^{a x}
| c =
}}
{{eqn | ll= \leadsto
| l = \frac {\d u} {\d x}
| r = a e^{a x}
| c = [[De... | Primitive of Exponential of a x over Power of x | https://proofwiki.org/wiki/Primitive_of_Exponential_of_a_x_over_Power_of_x | https://proofwiki.org/wiki/Primitive_of_Exponential_of_a_x_over_Power_of_x | [
"Primitives involving Exponential Function"
] | [] | [
"Definition:Primitive",
"Derivative of Exponential Function/Corollary 1",
"Primitive of Power",
"Integration by Parts",
"Primitive of Constant Multiple of Function"
] |
proofwiki-9721 | Primitive of Reciprocal of p plus q by Exponential of a x | :$\ds \int \frac {\d x} {p + q e^{a x} } = \frac x p - \frac 1 {a p} \ln \size {p + q e^{a x} } + C$ | {{begin-eqn}}
{{eqn | l = z
| r = p + q e^{a x}
| c =
}}
{{eqn | ll= \leadsto
| l = \frac {\d z} {\d x}
| r = a q e^{a x}
| c = Derivative of $e^{a x}$
}}
{{eqn | r = a \paren {z - p}
| c = in terms of $z$
}}
{{eqn | ll= \leadsto
| l = \int \frac {\d x} {p + q e^{a x} }
... | :$\ds \int \frac {\d x} {p + q e^{a x} } = \frac x p - \frac 1 {a p} \ln \size {p + q e^{a x} } + C$ | {{begin-eqn}}
{{eqn | l = z
| r = p + q e^{a x}
| c =
}}
{{eqn | ll= \leadsto
| l = \frac {\d z} {\d x}
| r = a q e^{a x}
| c = [[Derivative of Exponential of a x|Derivative of $e^{a x}$]]
}}
{{eqn | r = a \paren {z - p}
| c = in terms of $z$
}}
{{eqn | ll= \leadsto
| l = \int... | Primitive of Reciprocal of p plus q by Exponential of a x | https://proofwiki.org/wiki/Primitive_of_Reciprocal_of_p_plus_q_by_Exponential_of_a_x | https://proofwiki.org/wiki/Primitive_of_Reciprocal_of_p_plus_q_by_Exponential_of_a_x | [
"Primitives involving Exponential Function"
] | [] | [
"Derivative of Exponential Function/Corollary 1",
"Integration by Substitution",
"Primitive of Constant Multiple of Function",
"Primitive of Reciprocal of x by a x + b",
"Difference of Logarithms",
"Sum of Logarithms",
"Definition:Positive/Real Number",
"Definition:Primitive (Calculus)/Constant of Int... |
proofwiki-9722 | Primitive of Reciprocal of Square of p plus q by Exponential of a x | :$\ds \int \frac {\d x} {\paren {p + q e^{a x} }^2} = \frac x {p^2} + \frac 1 {a p \paren {p + q e^{a x} } } - \frac 1 {a p^2} \ln \size {p + q e^{a x} } + C$ | {{begin-eqn}}
{{eqn | l = z
| r = p + q e^{a x}
| c =
}}
{{eqn | ll= \leadsto
| l = \frac {\d z} {\d x}
| r = a q e^{a x}
| c = Derivative of $e^{a x}$
}}
{{eqn | r = a \paren {z - p}
| c = in terms of $z$
}}
{{eqn | ll= \leadsto
| l = \int \frac {\d x} {p + q e^{a x} }
... | :$\ds \int \frac {\d x} {\paren {p + q e^{a x} }^2} = \frac x {p^2} + \frac 1 {a p \paren {p + q e^{a x} } } - \frac 1 {a p^2} \ln \size {p + q e^{a x} } + C$ | {{begin-eqn}}
{{eqn | l = z
| r = p + q e^{a x}
| c =
}}
{{eqn | ll= \leadsto
| l = \frac {\d z} {\d x}
| r = a q e^{a x}
| c = [[Derivative of Exponential of a x|Derivative of $e^{a x}$]]
}}
{{eqn | r = a \paren {z - p}
| c = in terms of $z$
}}
{{eqn | ll= \leadsto
| l = \int... | Primitive of Reciprocal of Square of p plus q by Exponential of a x | https://proofwiki.org/wiki/Primitive_of_Reciprocal_of_Square_of_p_plus_q_by_Exponential_of_a_x | https://proofwiki.org/wiki/Primitive_of_Reciprocal_of_Square_of_p_plus_q_by_Exponential_of_a_x | [
"Primitives involving Exponential Function"
] | [] | [
"Derivative of Exponential Function/Corollary 1",
"Integration by Substitution",
"Primitive of Constant Multiple of Function",
"Primitive of Reciprocal of x squared by a x + b",
"Difference of Logarithms",
"Sum of Logarithms",
"Definition:Positive/Real Number",
"Definition:Primitive (Calculus)/Constan... |
proofwiki-9723 | Primitive of Reciprocal of p by Exponential of a x plus q by Exponential of -a x | :$\ds \int \frac {\d x} {p e^{a x} + q e^{-a x} } = \begin{cases}
\dfrac 1 {a \sqrt {p q} } \map \arctan {\sqrt {\dfrac p q} e^{a x} } & : \sqrt {p q} > 0 \\
\dfrac 1 {2 a \sqrt {-p q} } \ln \size {\dfrac {e^{a x} - \sqrt {-\dfrac q p} } {e^{a x} + \sqrt {-\dfrac q p} } } & : \sqrt {p q} < 0 \\
\end{cases}$ | {{begin-eqn}}
{{eqn | l = z
| r = e^{a x}
| c =
}}
{{eqn | ll= \leadsto
| l = \frac {\d z} {\d x}
| r = a e^{a x}
| c = Derivative of $e^{a x}$
}}
{{eqn | ll= \leadsto
| l = \int \frac {\d x} {p e^{a x} + q e^{-a x} }
| r = \int \frac 1 {p z + q z^{-1} } \frac {\d z} {a z}
... | :$\ds \int \frac {\d x} {p e^{a x} + q e^{-a x} } = \begin{cases}
\dfrac 1 {a \sqrt {p q} } \map \arctan {\sqrt {\dfrac p q} e^{a x} } & : \sqrt {p q} > 0 \\
\dfrac 1 {2 a \sqrt {-p q} } \ln \size {\dfrac {e^{a x} - \sqrt {-\dfrac q p} } {e^{a x} + \sqrt {-\dfrac q p} } } & : \sqrt {p q} < 0 \\
\end{cases}$ | {{begin-eqn}}
{{eqn | l = z
| r = e^{a x}
| c =
}}
{{eqn | ll= \leadsto
| l = \frac {\d z} {\d x}
| r = a e^{a x}
| c = [[Derivative of Exponential of a x|Derivative of $e^{a x}$]]
}}
{{eqn | ll= \leadsto
| l = \int \frac {\d x} {p e^{a x} + q e^{-a x} }
| r = \int \frac 1 {p ... | Primitive of Reciprocal of p by Exponential of a x plus q by Exponential of -a x | https://proofwiki.org/wiki/Primitive_of_Reciprocal_of_p_by_Exponential_of_a_x_plus_q_by_Exponential_of_-a_x | https://proofwiki.org/wiki/Primitive_of_Reciprocal_of_p_by_Exponential_of_a_x_plus_q_by_Exponential_of_-a_x | [
"Primitives involving Exponential Function"
] | [] | [
"Derivative of Exponential Function/Corollary 1",
"Integration by Substitution",
"Primitive of Constant Multiple of Function",
"Primitive of Reciprocal of x squared plus a squared/Arctangent Form",
"Primitive of Reciprocal of x squared minus a squared/Logarithm Form"
] |
proofwiki-9724 | Primitive of Exponential of a x by Sine of b x | :$\ds \int e^{a x} \sin b x \rd x = \frac {e^{a x} \paren {a \sin b x - b \cos b x} } {a^2 + b^2} + C$ | {{begin-eqn}}
{{eqn | l = \int e^{a x} \sin b x \rd x
| r = \frac {e^{a x} \sin b x} a - \frac b a \int e^{a x} \cos b x \rd x
| c = Primitive of $e^{a x} \sin b x$: Lemma
}}
{{eqn | r = \frac {e^{a x} \sin b x} a - \frac b a \paren {\frac {e^{a x} \cos b x} a + \frac b a \int e^{a x} \sin b x \rd x}
... | :$\ds \int e^{a x} \sin b x \rd x = \frac {e^{a x} \paren {a \sin b x - b \cos b x} } {a^2 + b^2} + C$ | {{begin-eqn}}
{{eqn | l = \int e^{a x} \sin b x \rd x
| r = \frac {e^{a x} \sin b x} a - \frac b a \int e^{a x} \cos b x \rd x
| c = [[Primitive of Exponential of a x by Sine of b x/Lemma|Primitive of $e^{a x} \sin b x$: Lemma]]
}}
{{eqn | r = \frac {e^{a x} \sin b x} a - \frac b a \paren {\frac {e^{a x} \c... | Primitive of Exponential of a x by Sine of b x/Proof 1 | https://proofwiki.org/wiki/Primitive_of_Exponential_of_a_x_by_Sine_of_b_x | https://proofwiki.org/wiki/Primitive_of_Exponential_of_a_x_by_Sine_of_b_x/Proof_1 | [
"Primitive of Exponential of a x by Sine of b x",
"Primitives involving Exponential Function",
"Primitives involving Sine Function"
] | [] | [
"Primitive of Exponential of a x by Sine of b x/Lemma",
"Primitive of Exponential of a x by Cosine of b x/Lemma",
"Definition:Common Denominator"
] |
proofwiki-9725 | Primitive of Exponential of a x by Sine of b x | :$\ds \int e^{a x} \sin b x \rd x = \frac {e^{a x} \paren {a \sin b x - b \cos b x} } {a^2 + b^2} + C$ | {{begin-eqn}}
{{eqn | l = \cos b x + i \sin b x
| r = e^{i b x}
| c = Euler's Formula
}}
{{eqn | ll= \leadsto
| l = e^{a x} \cos b x + i e^{a x} \sin b x
| r = e^{a x} e^{i b x}
| c = multiplying both sides by $e^{a x}$
}}
{{eqn | r = e^{\paren {a + i b} x}
| c = Exponent Combination... | :$\ds \int e^{a x} \sin b x \rd x = \frac {e^{a x} \paren {a \sin b x - b \cos b x} } {a^2 + b^2} + C$ | {{begin-eqn}}
{{eqn | l = \cos b x + i \sin b x
| r = e^{i b x}
| c = [[Euler's Formula]]
}}
{{eqn | ll= \leadsto
| l = e^{a x} \cos b x + i e^{a x} \sin b x
| r = e^{a x} e^{i b x}
| c = multiplying both sides by $e^{a x}$
}}
{{eqn | r = e^{\paren {a + i b} x}
| c = [[Exponent Combi... | Primitive of Exponential of a x by Sine of b x/Proof 2 | https://proofwiki.org/wiki/Primitive_of_Exponential_of_a_x_by_Sine_of_b_x | https://proofwiki.org/wiki/Primitive_of_Exponential_of_a_x_by_Sine_of_b_x/Proof_2 | [
"Primitive of Exponential of a x by Sine of b x",
"Primitives involving Exponential Function",
"Primitives involving Sine Function"
] | [] | [
"Euler's Formula",
"Exponent Combination Laws",
"Linear Combination of Complex Integrals",
"Primitive of Exponential of a x",
"Exponent Combination Laws",
"Euler's Formula"
] |
proofwiki-9726 | Primitive of Exponential of a x by Sine of b x | :$\ds \int e^{a x} \sin b x \rd x = \frac {e^{a x} \paren {a \sin b x - b \cos b x} } {a^2 + b^2} + C$ | {{begin-eqn}}
{{eqn | l = \int e^{a x} \sin b x \rd x
| r = \int e^{a x} \paren {\frac {e^{i b x} - e^{-i b x} } {2 i} } \rd x
| c = Euler's Sine Identity
}}
{{eqn | r = \frac 1 {2 i} \int e^{a x} \paren {e^{i b x} - e^{-i b x} } \rd x
| c = Primitive of Constant Multiple of Function
}}
{{eqn | r = \f... | :$\ds \int e^{a x} \sin b x \rd x = \frac {e^{a x} \paren {a \sin b x - b \cos b x} } {a^2 + b^2} + C$ | {{begin-eqn}}
{{eqn | l = \int e^{a x} \sin b x \rd x
| r = \int e^{a x} \paren {\frac {e^{i b x} - e^{-i b x} } {2 i} } \rd x
| c = [[Euler's Sine Identity]]
}}
{{eqn | r = \frac 1 {2 i} \int e^{a x} \paren {e^{i b x} - e^{-i b x} } \rd x
| c = [[Primitive of Constant Multiple of Function]]
}}
{{eqn ... | Primitive of Exponential of a x by Sine of b x/Proof 3 | https://proofwiki.org/wiki/Primitive_of_Exponential_of_a_x_by_Sine_of_b_x | https://proofwiki.org/wiki/Primitive_of_Exponential_of_a_x_by_Sine_of_b_x/Proof_3 | [
"Primitive of Exponential of a x by Sine of b x",
"Primitives involving Exponential Function",
"Primitives involving Sine Function"
] | [] | [
"Euler's Sine Identity",
"Primitive of Constant Multiple of Function",
"Exponent Combination Laws/Product of Powers",
"Linear Combination of Integrals/Indefinite",
"Primitive of Exponential of a x",
"Exponent Combination Laws/Product of Powers",
"Product of Complex Number with Conjugate",
"Euler's Cos... |
proofwiki-9727 | Primitive of Exponential of a x by Sine of b x | :$\ds \int e^{a x} \sin b x \rd x = \frac {e^{a x} \paren {a \sin b x - b \cos b x} } {a^2 + b^2} + C$ | Let $a, b, x \in \R$ be real numbers.
Suppose $a \ne 0 \ne b$.
Let $f_1$ and $f_2$ be the real functions defined as:
{{begin-eqn}}
{{eqn | q = \forall x \in \R
| l = \map {f_1} x
| r = \map \exp {a x} \map \cos {b x}
}}
{{eqn | l = \map {f_2} x
| r = \map \exp {a x} \map \sin {b x}
}}
{{end-eqn}}
Let ... | :$\ds \int e^{a x} \sin b x \rd x = \frac {e^{a x} \paren {a \sin b x - b \cos b x} } {a^2 + b^2} + C$ | Let $a, b, x \in \R$ be [[Definition:Real Number|real numbers]].
Suppose $a \ne 0 \ne b$.
Let $f_1$ and $f_2$ be the [[Definition:Real Function|real functions]] defined as:
{{begin-eqn}}
{{eqn | q = \forall x \in \R
| l = \map {f_1} x
| r = \map \exp {a x} \map \cos {b x}
}}
{{eqn | l = \map {f_2} x
... | Primitive of Exponential of a x by Sine of b x/Proof 4 | https://proofwiki.org/wiki/Primitive_of_Exponential_of_a_x_by_Sine_of_b_x | https://proofwiki.org/wiki/Primitive_of_Exponential_of_a_x_by_Sine_of_b_x/Proof_4 | [
"Primitive of Exponential of a x by Sine of b x",
"Primitives involving Exponential Function",
"Primitives involving Sine Function"
] | [] | [
"Definition:Real Number",
"Definition:Real Function",
"Definition:Continuous Real-Valued Function Space",
"Space of Continuously Differentiable on Closed Interval Real-Valued Functions with Pointwise Addition and Pointwise Scalar Multiplication forms Vector Space",
"Definition:Vector Space",
"Definition:D... |
proofwiki-9728 | Primitive of Exponential of a x by Cosine of b x | :$\ds \int e^{a x} \cos b x \rd x = \frac {e^{a x} \paren {a \cos b x + b \sin b x} } {a^2 + b^2} + C$ | {{begin-eqn}}
{{eqn | l = \int e^{a x} \cos b x \rd x
| r = \frac {e^{a x} \cos b x} a + \frac b a \int e^{a x} \sin b x \rd x
| c = Primitive of $e^{a x} \cos b x$: Lemma
}}
{{eqn | r = \frac {e^{a x} \cos b x} a + \frac b a \paren {\frac {e^{a x} \sin b x} a - \frac b a \int e^{a x} \cos b x \rd x}
... | :$\ds \int e^{a x} \cos b x \rd x = \frac {e^{a x} \paren {a \cos b x + b \sin b x} } {a^2 + b^2} + C$ | {{begin-eqn}}
{{eqn | l = \int e^{a x} \cos b x \rd x
| r = \frac {e^{a x} \cos b x} a + \frac b a \int e^{a x} \sin b x \rd x
| c = [[Primitive of Exponential of a x by Cosine of b x/Lemma|Primitive of $e^{a x} \cos b x$: Lemma]]
}}
{{eqn | r = \frac {e^{a x} \cos b x} a + \frac b a \paren {\frac {e^{a x} ... | Primitive of Exponential of a x by Cosine of b x/Proof 1 | https://proofwiki.org/wiki/Primitive_of_Exponential_of_a_x_by_Cosine_of_b_x | https://proofwiki.org/wiki/Primitive_of_Exponential_of_a_x_by_Cosine_of_b_x/Proof_1 | [
"Primitive of Exponential of a x by Cosine of b x",
"Primitives involving Exponential Function",
"Primitives involving Cosine Function"
] | [] | [
"Primitive of Exponential of a x by Cosine of b x/Lemma",
"Primitive of Exponential of a x by Sine of b x/Lemma"
] |
proofwiki-9729 | Primitive of Exponential of a x by Cosine of b x | :$\ds \int e^{a x} \cos b x \rd x = \frac {e^{a x} \paren {a \cos b x + b \sin b x} } {a^2 + b^2} + C$ | {{begin-eqn}}
{{eqn | l = \int e^{a x} e^{i b x} \rd x
| r = i \int e^{a x} \sin b x \rd x + \int e^{a x} \cos b x \rd x
| c = Euler's Formula
}}
{{eqn | ll= \leadsto
| l = \int e^{a x} \cos b x \rd x
| r = \map \Re {\int e^{\paren {a + i b} x} \rd x}
}}
{{eqn | r = \map \Re {\frac {e^{\paren {... | :$\ds \int e^{a x} \cos b x \rd x = \frac {e^{a x} \paren {a \cos b x + b \sin b x} } {a^2 + b^2} + C$ | {{begin-eqn}}
{{eqn | l = \int e^{a x} e^{i b x} \rd x
| r = i \int e^{a x} \sin b x \rd x + \int e^{a x} \cos b x \rd x
| c = [[Euler's Formula]]
}}
{{eqn | ll= \leadsto
| l = \int e^{a x} \cos b x \rd x
| r = \map \Re {\int e^{\paren {a + i b} x} \rd x}
}}
{{eqn | r = \map \Re {\frac {e^{\par... | Primitive of Exponential of a x by Cosine of b x/Proof 2 | https://proofwiki.org/wiki/Primitive_of_Exponential_of_a_x_by_Cosine_of_b_x | https://proofwiki.org/wiki/Primitive_of_Exponential_of_a_x_by_Cosine_of_b_x/Proof_2 | [
"Primitive of Exponential of a x by Cosine of b x",
"Primitives involving Exponential Function",
"Primitives involving Cosine Function"
] | [] | [
"Euler's Formula",
"Primitive of Exponential of a x",
"Euler's Formula",
"Definition:Complex Number/Real Part"
] |
proofwiki-9730 | Primitive of Exponential of a x by Cosine of b x | :$\ds \int e^{a x} \cos b x \rd x = \frac {e^{a x} \paren {a \cos b x + b \sin b x} } {a^2 + b^2} + C$ | Let $a, b \in \R_{>0}$ be real constants.
Let $f_1$ and $f_2$ be the real functions defined as:
{{begin-eqn}}
{{eqn | q = \forall x \in \R
| l = \map {f_1} x
| r = \map \exp {a x} \map \cos {b x}
}}
{{eqn | l = \map {f_2} x
| r = \map \exp {a x} \map \sin {b x}
}}
{{end-eqn}}
Let $\map \CC \R$ denote ... | :$\ds \int e^{a x} \cos b x \rd x = \frac {e^{a x} \paren {a \cos b x + b \sin b x} } {a^2 + b^2} + C$ | Let $a, b \in \R_{>0}$ be [[Definition:Real Number|real]] [[Definition:Constant|constants]].
Let $f_1$ and $f_2$ be the [[Definition:Real Function|real functions]] defined as:
{{begin-eqn}}
{{eqn | q = \forall x \in \R
| l = \map {f_1} x
| r = \map \exp {a x} \map \cos {b x}
}}
{{eqn | l = \map {f_2} x
... | Primitive of Exponential of a x by Cosine of b x/Proof 3 | https://proofwiki.org/wiki/Primitive_of_Exponential_of_a_x_by_Cosine_of_b_x | https://proofwiki.org/wiki/Primitive_of_Exponential_of_a_x_by_Cosine_of_b_x/Proof_3 | [
"Primitive of Exponential of a x by Cosine of b x",
"Primitives involving Exponential Function",
"Primitives involving Cosine Function"
] | [] | [
"Definition:Real Number",
"Definition:Constant",
"Definition:Real Function",
"Definition:Continuous Real-Valued Function Space",
"Space of Continuously Differentiable on Closed Interval Real-Valued Functions with Pointwise Addition and Pointwise Scalar Multiplication forms Vector Space",
"Definition:Vecto... |
proofwiki-9731 | Primitive of x by Exponential of a x by Sine of b x | :$\ds \int x e^{a x} \sin b x \rd x = \frac {x e^{a x} \paren {a \sin b x - b \cos b x} } {a^2 + b^2} - \frac {e^{a x} \paren {\paren {a^2 - b^2} \sin b x - 2 a b \cos b x} } {\paren {a^2 + b^2}^2} + C$ | 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 = x
| c =
}}
{{eqn | ll= \leadsto
| l = \frac {\d u} {\d x}
| r = 1
| c = Derivative of Identity Function
}}
{{end-eqn... | :$\ds \int x e^{a x} \sin b x \rd x = \frac {x e^{a x} \paren {a \sin b x - b \cos b x} } {a^2 + b^2} - \frac {e^{a x} \paren {\paren {a^2 - b^2} \sin b x - 2 a b \cos b x} } {\paren {a^2 + b^2}^2} + C$ | 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 = x
| c =
}}
{{eqn | ll= \leadsto
| l = \frac {\d u} {\d x}
| r = 1
| c = [[Derivative of Id... | Primitive of x by Exponential of a x by Sine of b x | https://proofwiki.org/wiki/Primitive_of_x_by_Exponential_of_a_x_by_Sine_of_b_x | https://proofwiki.org/wiki/Primitive_of_x_by_Exponential_of_a_x_by_Sine_of_b_x | [
"Primitives involving Exponential Function",
"Primitives involving Sine Function"
] | [] | [
"Definition:Primitive",
"Derivative of Identity Function",
"Primitive of Exponential of a x by Sine of b x",
"Integration by Parts",
"Linear Combination of Integrals/Indefinite",
"Primitive of Exponential of a x by Sine of b x",
"Primitive of Exponential of a x by Cosine of b x"
] |
proofwiki-9732 | Primitive of x by Exponential of a x by Cosine of b x | :$\ds \int x e^{a x} \cos b x \rd x = \frac {x e^{a x} \paren {a \cos b x + b \sin b x} } {a^2 + b^2} - \frac {e^{a x} \paren {\paren {a^2 - b^2} \cos b x - 2 a b \sin b x} } {\paren {a^2 + b^2}^2} + C$ | 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 = x
| c =
}}
{{eqn | ll= \leadsto
| l = \frac {\d u} {\d x}
| r = 1
| c = Derivative of Identity Function
}}
{{end-eqn... | :$\ds \int x e^{a x} \cos b x \rd x = \frac {x e^{a x} \paren {a \cos b x + b \sin b x} } {a^2 + b^2} - \frac {e^{a x} \paren {\paren {a^2 - b^2} \cos b x - 2 a b \sin b x} } {\paren {a^2 + b^2}^2} + C$ | 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 = x
| c =
}}
{{eqn | ll= \leadsto
| l = \frac {\d u} {\d x}
| r = 1
| c = [[Derivative of Id... | Primitive of x by Exponential of a x by Cosine of b x | https://proofwiki.org/wiki/Primitive_of_x_by_Exponential_of_a_x_by_Cosine_of_b_x | https://proofwiki.org/wiki/Primitive_of_x_by_Exponential_of_a_x_by_Cosine_of_b_x | [
"Primitives involving Exponential Function",
"Primitives involving Cosine Function"
] | [] | [
"Definition:Primitive",
"Derivative of Identity Function",
"Primitive of Exponential of a x by Cosine of b x",
"Integration by Parts",
"Linear Combination of Integrals/Indefinite",
"Primitive of Exponential of a x by Cosine of b x",
"Primitive of Exponential of a x by Sine of b x"
] |
proofwiki-9733 | Primitive of Exponential of a x by Logarithm of x | :$\ds \int e^{a x} \ln x \rd x = \frac {e^{a x} \ln x} a - \frac 1 a \int \frac {e^{a x} } x \rd x + C$ | 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 = \ln x
| c =
}}
{{eqn | ll= \leadsto
| l = \frac {\d u} {\d x}
| r = \frac 1 x
| c = Derivative of Natural Logarithm
... | :$\ds \int e^{a x} \ln x \rd x = \frac {e^{a x} \ln x} a - \frac 1 a \int \frac {e^{a x} } x \rd x + C$ | 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 = \ln x
| c =
}}
{{eqn | ll= \leadsto
| l = \frac {\d u} {\d x}
| r = \frac 1 x
| c = [[Deri... | Primitive of Exponential of a x by Logarithm of x | https://proofwiki.org/wiki/Primitive_of_Exponential_of_a_x_by_Logarithm_of_x | https://proofwiki.org/wiki/Primitive_of_Exponential_of_a_x_by_Logarithm_of_x | [
"Primitives involving Exponential Function",
"Primitives involving Logarithm Function"
] | [] | [
"Definition:Primitive",
"Derivative of Natural Logarithm Function",
"Primitive of Exponential of a x",
"Integration by Parts",
"Primitive of Constant Multiple of Function"
] |
proofwiki-9734 | Primitive of Exponential of a x by Power of Sine of b x | :$\ds \int e^{a x} \sin^n b x \rd x = \frac {e^{a x} \sin^{n - 1} b x} {a^2 + n^2 b^2} \paren {a \sin b x - n b \cos b x} + \frac {n \paren {n - 1} b^2} {a^2 + n^2 b^2} \int e^{a x} \sin^{n - 2} b x \rd x + C$ | 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 = \sin^n b x
| c =
}}
{{eqn | ll= \leadsto
| l = \frac {\d u} {\d x}
| r = n b \sin^{n - 1} b x \cos b x
| c = Derivat... | :$\ds \int e^{a x} \sin^n b x \rd x = \frac {e^{a x} \sin^{n - 1} b x} {a^2 + n^2 b^2} \paren {a \sin b x - n b \cos b x} + \frac {n \paren {n - 1} b^2} {a^2 + n^2 b^2} \int e^{a x} \sin^{n - 2} b x \rd x + C$ | 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 = \sin^n b x
| c =
}}
{{eqn | ll= \leadsto
| l = \frac {\d u} {\d x}
| r = n b \sin^{n - 1} b x \c... | Primitive of Exponential of a x by Power of Sine of b x | https://proofwiki.org/wiki/Primitive_of_Exponential_of_a_x_by_Power_of_Sine_of_b_x | https://proofwiki.org/wiki/Primitive_of_Exponential_of_a_x_by_Power_of_Sine_of_b_x | [
"Primitives involving Exponential Function",
"Primitives involving Sine Function",
"Primitive of Exponential of a x by Power of Sine of b x"
] | [] | [
"Definition:Primitive",
"Derivative of Sine Function/Corollary",
"Power Rule for Derivatives",
"Derivative of Composite Function",
"Primitive of Exponential of a x",
"Integration by Parts",
"Primitive of Constant Multiple of Function",
"Primitive of Exponential of a x by Power of Sine of b x/Lemma 1",... |
proofwiki-9735 | Primitive of Exponential of a x by Power of Cosine of b x | :$\ds \int e^{a x} \cos^n b x \rd x = \frac {e^{a x} \cos^{n - 1} b x} {a^2 + n^2 b^2} \paren {a \cos b x + n b \sin b x} + \frac {n \paren {n - 1} b^2} {a^2 + n^2 b^2} \int e^{a x} \cos^{n - 2} b x \rd x + C$ | 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 = \cos^n b x
| c =
}}
{{eqn | ll= \leadsto
| l = \frac {\d u} {\d x}
| r = -n b \cos^{n - 1} b x \sin b x
| c = Deriva... | :$\ds \int e^{a x} \cos^n b x \rd x = \frac {e^{a x} \cos^{n - 1} b x} {a^2 + n^2 b^2} \paren {a \cos b x + n b \sin b x} + \frac {n \paren {n - 1} b^2} {a^2 + n^2 b^2} \int e^{a x} \cos^{n - 2} b x \rd x + C$ | 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 = \cos^n b x
| c =
}}
{{eqn | ll= \leadsto
| l = \frac {\d u} {\d x}
| r = -n b \cos^{n - 1} b x \... | Primitive of Exponential of a x by Power of Cosine of b x | https://proofwiki.org/wiki/Primitive_of_Exponential_of_a_x_by_Power_of_Cosine_of_b_x | https://proofwiki.org/wiki/Primitive_of_Exponential_of_a_x_by_Power_of_Cosine_of_b_x | [
"Primitive of Exponential of a x by Power of Cosine of b x",
"Primitives involving Exponential Function",
"Primitives involving Cosine Function"
] | [] | [
"Definition:Primitive",
"Derivative of Cosine Function/Corollary",
"Power Rule for Derivatives",
"Derivative of Composite Function",
"Primitive of Exponential of a x",
"Integration by Parts",
"Primitive of Constant Multiple of Function",
"Primitive of Exponential of a x by Power of Cosine of b x/Lemma... |
proofwiki-9736 | Primitive of Logarithm of x | :$\ds \int \ln x \rd x = x \ln x - x + C$ | 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 = \ln x
| c =
}}
{{eqn | ll= \leadsto
| l = \frac {\d u} {\d x}
| r = \frac 1 x
| c = Derivative of $\ln x$
}}
{{end-e... | :$\ds \int \ln x \rd x = x \ln x - x + C$ | 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 = \ln x
| c =
}}
{{eqn | ll= \leadsto
| l = \frac {\d u} {\d x}
| r = \frac 1 x
| c = [[Deri... | Primitive of Logarithm of x/Proof 1 | https://proofwiki.org/wiki/Primitive_of_Logarithm_of_x | https://proofwiki.org/wiki/Primitive_of_Logarithm_of_x/Proof_1 | [
"Primitive of Logarithm of x",
"Primitives involving Logarithm Function"
] | [] | [
"Definition:Primitive",
"Derivative of Natural Logarithm Function",
"Primitive of Constant",
"Integration by Parts",
"Primitive of Constant"
] |
proofwiki-9737 | Primitive of Logarithm of x | :$\ds \int \ln x \rd x = x \ln x - x + C$ | Note that we have:
{{begin-eqn}}
{{eqn | l = \int_0^1 \ln x \rd x
| r = \int_0^1 x^0 \paren {\ln x}^1 \rd x
}}
{{eqn | r = \frac {\paren {-1}^1 \map \Gamma 2} {1^2}
| c = Definite Integral from $0$ to $1$ of $x^m \paren {\ln x}^n$
}}
{{eqn | r = -1
| c = Gamma Function Extends Factorial
}}
{{end-eqn}}
We therefore ... | :$\ds \int \ln x \rd x = x \ln x - x + C$ | Note that we have:
{{begin-eqn}}
{{eqn | l = \int_0^1 \ln x \rd x
| r = \int_0^1 x^0 \paren {\ln x}^1 \rd x
}}
{{eqn | r = \frac {\paren {-1}^1 \map \Gamma 2} {1^2}
| c = [[Definite Integral from 0 to 1 of Power of x by Power of Logarithm of x|Definite Integral from $0$ to $1$ of $x^m \paren {\ln x}^n$]]
}}
{{eqn |... | Primitive of Logarithm of x/Proof 2 | https://proofwiki.org/wiki/Primitive_of_Logarithm_of_x | https://proofwiki.org/wiki/Primitive_of_Logarithm_of_x/Proof_2 | [
"Primitive of Logarithm of x",
"Primitives involving Logarithm Function"
] | [] | [
"Definite Integral from 0 to 1 of Power of x by Power of Logarithm of x",
"Gamma Function Extends Factorial",
"Integration by Substitution",
"Difference of Logarithms",
"Primitive of Constant",
"Fundamental Theorem of Calculus",
"Definition:Real Number",
"Fundamental Theorem of Calculus/First Part",
... |
proofwiki-9738 | Primitive of x by Logarithm of x | :$\ds \int x \ln x \rd x = \frac {x^2} 2 \paren {\ln x - \frac 1 2} + C$ | {{begin-eqn}}
{{eqn | l = \int x \map \ln {x^2 + a^2} \rd x
| r = \frac {x^2 \map \ln {x^2 + a^2} } 2 - \int \frac {x^3} {x^2 + a^2} \rd x + C
| c = Primitive of $x^m \map \ln {x^2 + a^2}$ with $m = 1$
}}
{{eqn | r = \frac {x^2 \map \ln {x^2 + a^2} } 2 - \paren {\frac {x^2} 2 - \frac {a^2} 2 \map \ln {x^2 +... | :$\ds \int x \ln x \rd x = \frac {x^2} 2 \paren {\ln x - \frac 1 2} + C$ | {{begin-eqn}}
{{eqn | l = \int x \map \ln {x^2 + a^2} \rd x
| r = \frac {x^2 \map \ln {x^2 + a^2} } 2 - \int \frac {x^3} {x^2 + a^2} \rd x + C
| c = [[Primitive of Power of x by Logarithm of x squared plus a squared|Primitive of $x^m \map \ln {x^2 + a^2}$]] with $m = 1$
}}
{{eqn | r = \frac {x^2 \map \ln {x... | Primitive of x by Logarithm of x squared plus a squared/Proof 1 | https://proofwiki.org/wiki/Primitive_of_x_by_Logarithm_of_x | https://proofwiki.org/wiki/Primitive_of_x_by_Logarithm_of_x_squared_plus_a_squared/Proof_1 | [
"Primitive of x by Logarithm of x",
"Primitives involving Logarithm Function"
] | [] | [
"Primitive of Power of x by Logarithm of x squared plus a squared",
"Primitive of x cubed over x squared plus a squared"
] |
proofwiki-9739 | Primitive of x by Logarithm of x | :$\ds \int x \ln x \rd x = \frac {x^2} 2 \paren {\ln x - \frac 1 2} + C$ | 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 = \ln x
| c =
}}
{{eqn | ll= \leadsto
| l = \frac {\d u} {\d x}
| r = \frac 1 x
| c = Derivative of $\ln x$
}}
{{end-e... | :$\ds \int x \ln x \rd x = \frac {x^2} 2 \paren {\ln x - \frac 1 2} + C$ | With a view to expressing the [[Definition:Primitive (Calculus)|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 = \ln x
| c =
}}
{{eqn | ll= \leadsto
| l = \frac {\d u} {\d x}
| r = \frac 1 x
|... | Primitive of x by Logarithm of x/Proof 1 | https://proofwiki.org/wiki/Primitive_of_x_by_Logarithm_of_x | https://proofwiki.org/wiki/Primitive_of_x_by_Logarithm_of_x/Proof_1 | [
"Primitive of x by Logarithm of x",
"Primitives involving Logarithm Function"
] | [] | [
"Definition:Primitive (Calculus)",
"Derivative of Natural Logarithm Function",
"Primitive of Power",
"Integration by Parts",
"Primitive of Constant Multiple of Function",
"Primitive of Power"
] |
proofwiki-9740 | Primitive of x by Logarithm of x | :$\ds \int x \ln x \rd x = \frac {x^2} 2 \paren {\ln x - \frac 1 2} + C$ | From Primitive of $x^m \ln x$:
:$\ds \int x^m \ln x \rd x = \frac {x^{m + 1} } {m + 1} \paren {\ln x - \frac 1 {m + 1} } + C$
The result follows by setting $m = 1$.
{{qed}} | :$\ds \int x \ln x \rd x = \frac {x^2} 2 \paren {\ln x - \frac 1 2} + C$ | From [[Primitive of Power of x by Logarithm of x|Primitive of $x^m \ln x$]]:
:$\ds \int x^m \ln x \rd x = \frac {x^{m + 1} } {m + 1} \paren {\ln x - \frac 1 {m + 1} } + C$
The result follows by setting $m = 1$.
{{qed}} | Primitive of x by Logarithm of x/Proof 2 | https://proofwiki.org/wiki/Primitive_of_x_by_Logarithm_of_x | https://proofwiki.org/wiki/Primitive_of_x_by_Logarithm_of_x/Proof_2 | [
"Primitive of x by Logarithm of x",
"Primitives involving Logarithm Function"
] | [] | [
"Primitive of Power of x by Logarithm of x"
] |
proofwiki-9741 | Primitive of Power of x by Logarithm of x | :$\ds \int x^m \ln x \rd x = \frac {x^{m + 1} } {m + 1} \paren {\ln x - \frac 1 {m + 1} } + C$
where $m \ne -1$. | 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 = \ln x
| c =
}}
{{eqn | ll= \leadsto
| l = \frac {\d u} {\d x}
| r = \frac 1 x
| c = Derivative of $\ln x$
}}
{{end-e... | :$\ds \int x^m \ln x \rd x = \frac {x^{m + 1} } {m + 1} \paren {\ln x - \frac 1 {m + 1} } + C$
where $m \ne -1$. | 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 = \ln x
| c =
}}
{{eqn | ll= \leadsto
| l = \frac {\d u} {\d x}
| r = \frac 1 x
| c = [[Deri... | Primitive of Power of x by Logarithm of x | https://proofwiki.org/wiki/Primitive_of_Power_of_x_by_Logarithm_of_x | https://proofwiki.org/wiki/Primitive_of_Power_of_x_by_Logarithm_of_x | [
"Primitives involving Logarithm Function"
] | [] | [
"Definition:Primitive",
"Derivative of Natural Logarithm Function",
"Primitive of Power",
"Integration by Parts",
"Primitive of Constant Multiple of Function",
"Primitive of Power"
] |
proofwiki-9742 | Primitive of Logarithm of x over x | :$\ds \int \frac {\ln x} x \rd x = \frac {\ln^2 x} 2 + C$ | 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 = \ln x
| c =
}}
{{eqn | ll= \leadsto
| l = \frac {\d u} {\d x}
| r = \frac 1 x
| c = Derivative of $\ln x$
}}
{{end-e... | :$\ds \int \frac {\ln x} x \rd x = \frac {\ln^2 x} 2 + C$ | 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 = \ln x
| c =
}}
{{eqn | ll= \leadsto
| l = \frac {\d u} {\d x}
| r = \frac 1 x
| c = [[Deri... | Primitive of Logarithm of x over x | https://proofwiki.org/wiki/Primitive_of_Logarithm_of_x_over_x | https://proofwiki.org/wiki/Primitive_of_Logarithm_of_x_over_x | [
"Primitives involving Logarithm Function"
] | [] | [
"Definition:Primitive",
"Derivative of Natural Logarithm Function",
"Primitive of Reciprocal",
"Integration by Parts"
] |
proofwiki-9743 | Primitive of Logarithm of x over x squared | :$\ds \int \frac {\ln x} {x^2} \rd x = \frac {-\ln x} x - \frac 1 x + C$ | From Primitive of $x^m \ln x$:
:$\ds \int x^m \ln x \rd x = \frac {x^{m + 1} } {m + 1} \paren {\ln x - \frac 1 {m + 1} } + C$
Thus:
{{begin-eqn}}
{{eqn | l = \int \frac {\ln x} {x^2} \rd x
| r = \frac {x^{-1} } {-1} \paren {\ln x - \frac 1 {-1} } + C
| c = Primitive of $x^m \ln x$, setting $m = -2$
}}
{{eqn... | :$\ds \int \frac {\ln x} {x^2} \rd x = \frac {-\ln x} x - \frac 1 x + C$ | From [[Primitive of Power of x by Logarithm of x|Primitive of $x^m \ln x$]]:
:$\ds \int x^m \ln x \rd x = \frac {x^{m + 1} } {m + 1} \paren {\ln x - \frac 1 {m + 1} } + C$
Thus:
{{begin-eqn}}
{{eqn | l = \int \frac {\ln x} {x^2} \rd x
| r = \frac {x^{-1} } {-1} \paren {\ln x - \frac 1 {-1} } + C
| c = [[P... | Primitive of Logarithm of x over x squared | https://proofwiki.org/wiki/Primitive_of_Logarithm_of_x_over_x_squared | https://proofwiki.org/wiki/Primitive_of_Logarithm_of_x_over_x_squared | [
"Primitives involving Logarithm Function"
] | [] | [
"Primitive of Power of x by Logarithm of x",
"Primitive of Power of x by Logarithm of x"
] |
proofwiki-9744 | Primitive of Square of Logarithm of x | :$\ds \int \ln^2 x \rd x = x \ln^2 x - 2 x \ln x + 2 x + C$ | 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 = \ln x
| c =
}}
{{eqn | ll= \leadsto
| l = \frac {\d u} {\d x}
| r = \frac 1 x
| c = Derivative of $\ln x$
}}
{{end-e... | :$\ds \int \ln^2 x \rd x = x \ln^2 x - 2 x \ln x + 2 x + C$ | 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 = \ln x
| c =
}}
{{eqn | ll= \leadsto
| l = \frac {\d u} {\d x}
| r = \frac 1 x
| c = [[Deri... | Primitive of Square of Logarithm of x | https://proofwiki.org/wiki/Primitive_of_Square_of_Logarithm_of_x | https://proofwiki.org/wiki/Primitive_of_Square_of_Logarithm_of_x | [
"Primitives involving Logarithm Function"
] | [] | [
"Definition:Primitive",
"Derivative of Natural Logarithm Function",
"Primitive of Logarithm of x",
"Integration by Parts",
"Linear Combination of Integrals/Indefinite",
"Primitive of Logarithm of x",
"Primitive of Constant"
] |
proofwiki-9745 | Primitive of Power of Logarithm of x over x | :$\ds \int \frac {\ln^n x} x \rd x = \frac {\ln^{n + 1} x} {n + 1} + C$ | {{begin-eqn}}
{{eqn | l = z
| r = \ln x
| c =
}}
{{eqn | ll= \leadsto
| l = \frac {\d z} {\d x}
| r = \frac 1 x
| c = Derivative of Natural Logarithm
}}
{{eqn | ll= \leadsto
| l = \int \frac {\ln^n x} x \rd x
| r = \int z^n \rd z
| c = Integration by Substitution
}}
{{eq... | :$\ds \int \frac {\ln^n x} x \rd x = \frac {\ln^{n + 1} x} {n + 1} + C$ | {{begin-eqn}}
{{eqn | l = z
| r = \ln x
| c =
}}
{{eqn | ll= \leadsto
| l = \frac {\d z} {\d x}
| r = \frac 1 x
| c = [[Derivative of Natural Logarithm]]
}}
{{eqn | ll= \leadsto
| l = \int \frac {\ln^n x} x \rd x
| r = \int z^n \rd z
| c = [[Integration by Substitution]]... | Primitive of Power of Logarithm of x over x | https://proofwiki.org/wiki/Primitive_of_Power_of_Logarithm_of_x_over_x | https://proofwiki.org/wiki/Primitive_of_Power_of_Logarithm_of_x_over_x | [
"Primitives involving Logarithm Function"
] | [] | [
"Derivative of Natural Logarithm Function",
"Integration by Substitution",
"Primitive of Power"
] |
proofwiki-9746 | Primitive of Reciprocal of x by Logarithm of x | :$\ds \int \frac {\d x} {x \ln x} = \ln \size {\ln x} + C$ | {{begin-eqn}}
{{eqn | l = z
| r = \ln x
| c =
}}
{{eqn | ll= \leadsto
| l = \frac {\d z} {\d x}
| r = \frac 1 x
| c = Derivative of Natural Logarithm
}}
{{eqn | ll= \leadsto
| l = \int \frac {\d x} {x \ln x}
| r = \ln \size {\ln x} + C
| c = Primitive of Function under i... | :$\ds \int \frac {\d x} {x \ln x} = \ln \size {\ln x} + C$ | {{begin-eqn}}
{{eqn | l = z
| r = \ln x
| c =
}}
{{eqn | ll= \leadsto
| l = \frac {\d z} {\d x}
| r = \frac 1 x
| c = [[Derivative of Natural Logarithm]]
}}
{{eqn | ll= \leadsto
| l = \int \frac {\d x} {x \ln x}
| r = \ln \size {\ln x} + C
| c = [[Primitive of Function u... | Primitive of Reciprocal of x by Logarithm of x | https://proofwiki.org/wiki/Primitive_of_Reciprocal_of_x_by_Logarithm_of_x | https://proofwiki.org/wiki/Primitive_of_Reciprocal_of_x_by_Logarithm_of_x | [
"Primitives involving Logarithm Function"
] | [] | [
"Derivative of Natural Logarithm Function",
"Primitive of Function under its Derivative"
] |
proofwiki-9747 | Primitive of Reciprocal of Logarithm of x | For $x > 1$:
{{begin-eqn}}
{{eqn | l = \int \frac {\d x} {\ln x}
| r = \map \ln {\ln x} + \sum_{k \mathop \ge 1} \frac {\paren {\ln x}^k} {k \times k!} + C
| c =
}}
{{eqn | r = \map \ln {\ln x} + \dfrac {\ln x} {1 \times 1!} + \dfrac {\paren {\ln x}^2} {2 \times 2!} + \dfrac {\paren {\ln x}^3} {3 \times 3!... | From Primitive of $\dfrac {x^m} {\ln x}$:
:$\ds \int \frac {x^m \rd x} {\ln x} = \map \ln {\ln x} + \paren {m + 1} \ln x + \sum_{k \mathop \ge 2}^n \frac {\paren {m + 1}^k \paren {\ln x}^k} {k \times k!} + C$
The result follows by setting $m = 0$.
{{qed}} | For $x > 1$:
{{begin-eqn}}
{{eqn | l = \int \frac {\d x} {\ln x}
| r = \map \ln {\ln x} + \sum_{k \mathop \ge 1} \frac {\paren {\ln x}^k} {k \times k!} + C
| c =
}}
{{eqn | r = \map \ln {\ln x} + \dfrac {\ln x} {1 \times 1!} + \dfrac {\paren {\ln x}^2} {2 \times 2!} + \dfrac {\paren {\ln x}^3} {3 \times 3... | From [[Primitive of Power of x over Logarithm of x|Primitive of $\dfrac {x^m} {\ln x}$]]:
:$\ds \int \frac {x^m \rd x} {\ln x} = \map \ln {\ln x} + \paren {m + 1} \ln x + \sum_{k \mathop \ge 2}^n \frac {\paren {m + 1}^k \paren {\ln x}^k} {k \times k!} + C$
The result follows by setting $m = 0$.
{{qed}} | Primitive of Reciprocal of Logarithm of x/Proof 2 | https://proofwiki.org/wiki/Primitive_of_Reciprocal_of_Logarithm_of_x | https://proofwiki.org/wiki/Primitive_of_Reciprocal_of_Logarithm_of_x/Proof_2 | [
"Primitives involving Logarithm Function",
"Primitive of Reciprocal of Logarithm of x"
] | [] | [
"Primitive of Power of x over Logarithm of x"
] |
proofwiki-9748 | Primitive of Power of x over Logarithm of x | For $x > 1$:
{{begin-eqn}}
{{eqn | l = \int \frac {x^m \rd x} {\ln x}
| r = \map \ln {\ln x} + \sum_{k \mathop \ge 1} \frac {\paren {m + 1}^k \paren {\ln x}^k} {k \times k!} + C
| c =
}}
{{eqn | r = \map \ln {\ln x} + \dfrac {\paren {m + 1} \ln x} {1 \times 1!} + \frac {\paren {m + 1}^2 \paren {\ln x}^2} {... | {{begin-eqn}}
{{eqn | l = u
| r = \ln x
| c =
}}
{{eqn | ll= \leadsto
| l = x
| r = e^u
| c =
}}
{{eqn | ll= \leadsto
| l = \d x
| r = e^u \d u
| c =
}}
{{eqn | ll= \leadsto
| l = \int \frac {x^m \rd x} {\ln x}
| r = \int \frac {\paren {e^u}^m e^u \rd u} u
... | For $x > 1$:
{{begin-eqn}}
{{eqn | l = \int \frac {x^m \rd x} {\ln x}
| r = \map \ln {\ln x} + \sum_{k \mathop \ge 1} \frac {\paren {m + 1}^k \paren {\ln x}^k} {k \times k!} + C
| c =
}}
{{eqn | r = \map \ln {\ln x} + \dfrac {\paren {m + 1} \ln x} {1 \times 1!} + \frac {\paren {m + 1}^2 \paren {\ln x}^2} ... | {{begin-eqn}}
{{eqn | l = u
| r = \ln x
| c =
}}
{{eqn | ll= \leadsto
| l = x
| r = e^u
| c =
}}
{{eqn | ll= \leadsto
| l = \d x
| r = e^u \d u
| c =
}}
{{eqn | ll= \leadsto
| l = \int \frac {x^m \rd x} {\ln x}
| r = \int \frac {\paren {e^u}^m e^u \rd u} u
... | Primitive of Power of x over Logarithm of x | https://proofwiki.org/wiki/Primitive_of_Power_of_x_over_Logarithm_of_x | https://proofwiki.org/wiki/Primitive_of_Power_of_x_over_Logarithm_of_x | [
"Primitives involving Logarithm Function"
] | [] | [
"Primitive of Exponential of a x over x"
] |
proofwiki-9749 | Primitive of Power of Logarithm of x | :$\ds \int \ln^n x \rd x = x \ln^n x - n \int \ln^{n - 1} x \rd x + C$ | 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 = \ln^n x
| c =
}}
{{eqn | ll= \leadsto
| l = \frac {\d u} {\d x}
| r = n \ln^{n - 1} x \frac 1 x
| c = Derivative of ... | :$\ds \int \ln^n x \rd x = x \ln^n x - n \int \ln^{n - 1} x \rd x + C$ | 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 = \ln^n x
| c =
}}
{{eqn | ll= \leadsto
| l = \frac {\d u} {\d x}
| r = n \ln^{n - 1} x \frac 1 x
... | Primitive of Power of Logarithm of x | https://proofwiki.org/wiki/Primitive_of_Power_of_Logarithm_of_x | https://proofwiki.org/wiki/Primitive_of_Power_of_Logarithm_of_x | [
"Primitives involving Logarithm Function"
] | [] | [
"Definition:Primitive",
"Derivative of Natural Logarithm Function",
"Power Rule for Derivatives",
"Derivative of Composite Function",
"Primitive of Constant",
"Integration by Parts",
"Primitive of Constant Multiple of Function"
] |
proofwiki-9750 | Primitive of Power of x by Power of Logarithm of x | :$\ds \int x^m \ln^n x \rd x = \frac {x^{m + 1} \ln^n x} {m + 1} - \frac n {m + 1} \int x^m \ln^{n - 1} x \rd x + C$
where $m \ne -1$. | 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 = \ln^n x
| c =
}}
{{eqn | ll= \leadsto
| l = \frac {\d u} {\d x}
| r = n \ln^{n - 1} x \frac 1 x
| c = Derivative of ... | :$\ds \int x^m \ln^n x \rd x = \frac {x^{m + 1} \ln^n x} {m + 1} - \frac n {m + 1} \int x^m \ln^{n - 1} x \rd x + C$
where $m \ne -1$. | 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 = \ln^n x
| c =
}}
{{eqn | ll= \leadsto
| l = \frac {\d u} {\d x}
| r = n \ln^{n - 1} x \frac 1 x
... | Primitive of Power of x by Power of Logarithm of x | https://proofwiki.org/wiki/Primitive_of_Power_of_x_by_Power_of_Logarithm_of_x | https://proofwiki.org/wiki/Primitive_of_Power_of_x_by_Power_of_Logarithm_of_x | [
"Primitives involving Logarithm Function"
] | [] | [
"Definition:Primitive",
"Derivative of Natural Logarithm Function",
"Power Rule for Derivatives",
"Derivative of Composite Function",
"Primitive of Power",
"Integration by Parts",
"Primitive of Constant Multiple of Function"
] |
proofwiki-9751 | Primitive of Logarithm of x squared plus a squared | :$\ds \int \map \ln {x^2 + a^2} \rd x = x \map \ln {x^2 + a^2} - 2 x + 2 a \arctan \frac x a + C$ | 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 = \map \ln {x^2 + a^2}
| c =
}}
{{eqn | ll= \leadsto
| l = \frac {\d u} {\d x}
| r = \frac {2 x} {x^2 + a^2}
| c = Der... | :$\ds \int \map \ln {x^2 + a^2} \rd x = x \map \ln {x^2 + a^2} - 2 x + 2 a \arctan \frac x a + C$ | 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 = \map \ln {x^2 + a^2}
| c =
}}
{{eqn | ll= \leadsto
| l = \frac {\d u} {\d x}
| r = \frac {2 x} {... | Primitive of Logarithm of x squared plus a squared | https://proofwiki.org/wiki/Primitive_of_Logarithm_of_x_squared_plus_a_squared | https://proofwiki.org/wiki/Primitive_of_Logarithm_of_x_squared_plus_a_squared | [
"Primitives involving Logarithm Function",
"Primitives involving x squared plus a squared"
] | [] | [
"Definition:Primitive",
"Derivative of Natural Logarithm Function",
"Power Rule for Derivatives",
"Derivative of Composite Function",
"Primitive of Power",
"Integration by Parts",
"Primitive of Constant Multiple of Function",
"Primitive of x squared over x squared plus a squared"
] |
proofwiki-9752 | Primitive of Logarithm of x squared minus a squared | :$\ds \int \map \ln {x^2 - a^2} \rd x = x \map \ln {x^2 - a^2} - 2 x + a \map \ln {\frac {x + a} {x - a} } + C$
for $x^2 > a^2$. | 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 = \map \ln {x^2 - a^2}
| c =
}}
{{eqn | ll= \leadsto
| l = \frac {\d u} {\d x}
| r = \frac {2 x} {x^2 - a^2}
| c = Der... | :$\ds \int \map \ln {x^2 - a^2} \rd x = x \map \ln {x^2 - a^2} - 2 x + a \map \ln {\frac {x + a} {x - a} } + C$
for $x^2 > a^2$. | 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 = \map \ln {x^2 - a^2}
| c =
}}
{{eqn | ll= \leadsto
| l = \frac {\d u} {\d x}
| r = \frac {2 x} {... | Primitive of Logarithm of x squared minus a squared | https://proofwiki.org/wiki/Primitive_of_Logarithm_of_x_squared_minus_a_squared | https://proofwiki.org/wiki/Primitive_of_Logarithm_of_x_squared_minus_a_squared | [
"Primitives involving Logarithm Function",
"Primitives involving x squared minus a squared"
] | [] | [
"Definition:Primitive",
"Derivative of Natural Logarithm Function",
"Power Rule for Derivatives",
"Derivative of Composite Function",
"Primitive of Power",
"Integration by Parts",
"Primitive of Constant Multiple of Function",
"Primitive of x squared over x squared minus a squared",
"Logarithm of Rec... |
proofwiki-9753 | Primitive of Power of x by Logarithm of x squared plus a squared | :$\ds \int x^m \map \ln {x^2 + a^2} \rd x = \frac {x^{m + 1} \map \ln {x^2 + a^2} } {m + 1} - \frac 2 {m + 1} \int \frac {x^{m + 2} } {x^2 + a^2} \rd x$ | 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 = \map \ln {x^2 + a^2}
| c =
}}
{{eqn | ll= \leadsto
| l = \frac {\d u} {\d x}
| r = \frac {2 x} {x^2 + a^2}
| c = Der... | :$\ds \int x^m \map \ln {x^2 + a^2} \rd x = \frac {x^{m + 1} \map \ln {x^2 + a^2} } {m + 1} - \frac 2 {m + 1} \int \frac {x^{m + 2} } {x^2 + a^2} \rd x$ | 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 = \map \ln {x^2 + a^2}
| c =
}}
{{eqn | ll= \leadsto
| l = \frac {\d u} {\d x}
| r = \frac {2 x} {... | Primitive of Power of x by Logarithm of x squared plus a squared | https://proofwiki.org/wiki/Primitive_of_Power_of_x_by_Logarithm_of_x_squared_plus_a_squared | https://proofwiki.org/wiki/Primitive_of_Power_of_x_by_Logarithm_of_x_squared_plus_a_squared | [
"Primitives involving Logarithm Function",
"Primitives involving x squared plus a squared"
] | [] | [
"Definition:Primitive",
"Derivative of Natural Logarithm Function",
"Power Rule for Derivatives",
"Derivative of Composite Function",
"Primitive of Power",
"Integration by Parts",
"Primitive of Constant Multiple of Function"
] |
proofwiki-9754 | Primitive of Power of x by Logarithm of x squared minus a squared | :$\ds \int x^m \, \map \ln {x^2 - a^2} \rd x = \frac {x^{m + 1} \map \ln {x^2 - a^2} } {m + 1} - \frac 2 {m + 1} \int \frac {x^{m + 2} } {x^2 - a^2} \rd x$ | With a view to expressing the primitive in the form:
:$\ds \int u \frac {\d v} {\d x} \rd x = u v - \int v \frac {\rd u} {\rd x} \rd x$
let:
{{begin-eqn}}
{{eqn | l = u
| r = \map \ln {x^2 - a^2}
| c =
}}
{{eqn | ll= \leadsto
| l = \frac {\d u} {\d x}
| r = \frac {2 x} {x^2 - a^2}
| c = D... | :$\ds \int x^m \, \map \ln {x^2 - a^2} \rd x = \frac {x^{m + 1} \map \ln {x^2 - a^2} } {m + 1} - \frac 2 {m + 1} \int \frac {x^{m + 2} } {x^2 - a^2} \rd x$ | With a view to expressing the [[Definition:Primitive (Calculus)|primitive]] in the form:
:$\ds \int u \frac {\d v} {\d x} \rd x = u v - \int v \frac {\rd u} {\rd x} \rd x$
let:
{{begin-eqn}}
{{eqn | l = u
| r = \map \ln {x^2 - a^2}
| c =
}}
{{eqn | ll= \leadsto
| l = \frac {\d u} {\d x}
| r = ... | Primitive of Power of x by Logarithm of x squared minus a squared | https://proofwiki.org/wiki/Primitive_of_Power_of_x_by_Logarithm_of_x_squared_minus_a_squared | https://proofwiki.org/wiki/Primitive_of_Power_of_x_by_Logarithm_of_x_squared_minus_a_squared | [
"Primitives involving Logarithm Function",
"Primitives involving x squared minus a squared"
] | [] | [
"Definition:Primitive (Calculus)",
"Derivative of Natural Logarithm Function",
"Power Rule for Derivatives",
"Derivative of Composite Function",
"Primitive of Power",
"Integration by Parts",
"Primitive of Constant Multiple of Function"
] |
proofwiki-9755 | Primitive of Hyperbolic Sine of a x | :$\ds \int \sinh a x \rd x = \frac {\cosh a x} a + C$ | {{begin-eqn}}
{{eqn | l = \int \sinh x \rd x
| r = \cosh x + C
| c = Primitive of $\sinh x$
}}
{{eqn | ll= \leadsto
| l = \int \sinh a x \rd x
| r = \frac 1 a \paren {\cosh a x} + C
| c = Primitive of Function of Constant Multiple
}}
{{eqn | r = \frac {\cosh a x} a + C
| c = simplify... | :$\ds \int \sinh a x \rd x = \frac {\cosh a x} a + C$ | {{begin-eqn}}
{{eqn | l = \int \sinh x \rd x
| r = \cosh x + C
| c = [[Primitive of Hyperbolic Sine Function|Primitive of $\sinh x$]]
}}
{{eqn | ll= \leadsto
| l = \int \sinh a x \rd x
| r = \frac 1 a \paren {\cosh a x} + C
| c = [[Primitive of Function of Constant Multiple]]
}}
{{eqn | r ... | Primitive of Hyperbolic Sine of a x | https://proofwiki.org/wiki/Primitive_of_Hyperbolic_Sine_of_a_x | https://proofwiki.org/wiki/Primitive_of_Hyperbolic_Sine_of_a_x | [
"Primitives involving Hyperbolic Sine Function"
] | [] | [
"Primitive of Hyperbolic Sine Function",
"Primitive of Function of Constant Multiple"
] |
proofwiki-9756 | Primitive of Hyperbolic Sine of a x | :$\ds \int \sinh a x \rd x = \frac {\cosh a x} a + C$ | {{begin-eqn}}
{{eqn | l = \int \sinh a x \cosh a x \rd x
| r = \int \frac {\sinh 2 a x} 2 \rd x
| c = Double Angle Formula for Hyperbolic Sine
}}
{{eqn | r = \frac 1 2 \int \sinh 2 a x \rd x
| c = Primitive of Constant Multiple of Function
}}
{{eqn | r = \frac 1 2 \paren {\frac {\cosh 2 a x} {2 a} } +... | :$\ds \int \sinh a x \rd x = \frac {\cosh a x} a + C$ | {{begin-eqn}}
{{eqn | l = \int \sinh a x \cosh a x \rd x
| r = \int \frac {\sinh 2 a x} 2 \rd x
| c = [[Double Angle Formula for Hyperbolic Sine]]
}}
{{eqn | r = \frac 1 2 \int \sinh 2 a x \rd x
| c = [[Primitive of Constant Multiple of Function]]
}}
{{eqn | r = \frac 1 2 \paren {\frac {\cosh 2 a x} {... | Primitive of Hyperbolic Sine of a x by Hyperbolic Cosine of a x/Proof 1 | https://proofwiki.org/wiki/Primitive_of_Hyperbolic_Sine_of_a_x | https://proofwiki.org/wiki/Primitive_of_Hyperbolic_Sine_of_a_x_by_Hyperbolic_Cosine_of_a_x/Proof_1 | [
"Primitives involving Hyperbolic Sine Function"
] | [] | [
"Double Angle Formulas/Hyperbolic Sine",
"Primitive of Constant Multiple of Function",
"Primitive of Hyperbolic Sine of a x",
"Definition:Primitive (Calculus)/Constant of Integration"
] |
proofwiki-9757 | Primitive of Hyperbolic Sine of a x | :$\ds \int \sinh a x \rd x = \frac {\cosh a x} a + C$ | {{begin-eqn}}
{{eqn | l = \int \sinh a x \cosh a x \rd x
| r = \int \cosh a x \sinh a x \rd x
| c =
}}
{{eqn | r = \frac {\cosh^2 a x} {2 a} + C
| c = Primitive of $\cosh^n a x \sinh a x$ using $n = 1$
}}
{{eqn | r = \frac {1 + \sinh^2 a x} {2 a} + C
| c = Difference of Squares of Hyperbolic Co... | :$\ds \int \sinh a x \rd x = \frac {\cosh a x} a + C$ | {{begin-eqn}}
{{eqn | l = \int \sinh a x \cosh a x \rd x
| r = \int \cosh a x \sinh a x \rd x
| c =
}}
{{eqn | r = \frac {\cosh^2 a x} {2 a} + C
| c = [[Primitive of Power of Hyperbolic Cosine of a x by Hyperbolic Sine of a x|Primitive of $\cosh^n a x \sinh a x$]] using $n = 1$
}}
{{eqn | r = \frac {... | Primitive of Hyperbolic Sine of a x by Hyperbolic Cosine of a x/Proof 2 | https://proofwiki.org/wiki/Primitive_of_Hyperbolic_Sine_of_a_x | https://proofwiki.org/wiki/Primitive_of_Hyperbolic_Sine_of_a_x_by_Hyperbolic_Cosine_of_a_x/Proof_2 | [
"Primitives involving Hyperbolic Sine Function"
] | [] | [
"Primitive of Power of Hyperbolic Cosine of a x by Hyperbolic Sine of a x",
"Difference of Squares of Hyperbolic Cosine and Sine",
"Definition:Primitive (Calculus)/Constant of Integration"
] |
proofwiki-9758 | Primitive of Hyperbolic Sine of a x | :$\ds \int \sinh a x \rd x = \frac {\cosh a x} a + C$ | {{begin-eqn}}
{{eqn | n = 1
| l = \int \sinh^n a x \cosh a x \rd x
| r = \frac {\sinh^{n + 1} a x} {\paren {n + 1} a} + C
| c = Primitive of $\sinh^n a x \cosh a x$
}}
{{eqn | ll= \leadsto
| l = \int \sinh a x \cosh a x \rd x
| r = \frac {\sinh^2 a x} {2 a} + C
| c = setting $n = 1$ ... | :$\ds \int \sinh a x \rd x = \frac {\cosh a x} a + C$ | {{begin-eqn}}
{{eqn | n = 1
| l = \int \sinh^n a x \cosh a x \rd x
| r = \frac {\sinh^{n + 1} a x} {\paren {n + 1} a} + C
| c = [[Primitive of Power of Hyperbolic Sine of a x by Hyperbolic Cosine of a x|Primitive of $\sinh^n a x \cosh a x$]]
}}
{{eqn | ll= \leadsto
| l = \int \sinh a x \cosh a x... | Primitive of Hyperbolic Sine of a x by Hyperbolic Cosine of a x/Proof 3 | https://proofwiki.org/wiki/Primitive_of_Hyperbolic_Sine_of_a_x | https://proofwiki.org/wiki/Primitive_of_Hyperbolic_Sine_of_a_x_by_Hyperbolic_Cosine_of_a_x/Proof_3 | [
"Primitives involving Hyperbolic Sine Function"
] | [] | [
"Primitive of Power of Hyperbolic Sine of a x by Hyperbolic Cosine of a x"
] |
proofwiki-9759 | Primitive of Hyperbolic Sine of a x | :$\ds \int \sinh a x \rd x = \frac {\cosh a x} a + C$ | 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 = \sinh a x
| c =
}}
{{eqn | ll= \leadsto
| l = \frac {\d u} {\d x}
| r = a \cosh a x
| c = Derivative of $\sinh a x$
... | :$\ds \int \sinh a x \rd x = \frac {\cosh a x} a + C$ | With a view to expressing the [[Definition:Primitive (Calculus)|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 = \sinh a x
| c =
}}
{{eqn | ll= \leadsto
| l = \frac {\d u} {\d x}
| r = a \cosh a x
... | Primitive of Hyperbolic Sine of a x by Hyperbolic Cosine of a x/Proof 4 | https://proofwiki.org/wiki/Primitive_of_Hyperbolic_Sine_of_a_x | https://proofwiki.org/wiki/Primitive_of_Hyperbolic_Sine_of_a_x_by_Hyperbolic_Cosine_of_a_x/Proof_4 | [
"Primitives involving Hyperbolic Sine Function"
] | [] | [
"Definition:Primitive (Calculus)",
"Derivative of Hyperbolic Sine of a x",
"Primitive of Hyperbolic Cosine of a x",
"Integration by Parts"
] |
proofwiki-9760 | Primitive of Hyperbolic Sine of a x | :$\ds \int \sinh a x \rd x = \frac {\cosh a x} a + C$ | {{begin-eqn}}
{{eqn | l = u
| r = \sinh a x
| c =
}}
{{eqn | ll= \leadsto
| l = \frac {\d u} {\d x}
| r = a \cosh a x
| c = Derivative of $\sinh a x$
}}
{{eqn | ll= \leadsto
| l = \int \sinh a x \cosh a x \rd x
| r = \int \frac u a \rd u
| c = Integration by Substitution... | :$\ds \int \sinh a x \rd x = \frac {\cosh a x} a + C$ | {{begin-eqn}}
{{eqn | l = u
| r = \sinh a x
| c =
}}
{{eqn | ll= \leadsto
| l = \frac {\d u} {\d x}
| r = a \cosh a x
| c = [[Derivative of Hyperbolic Sine of a x|Derivative of $\sinh a x$]]
}}
{{eqn | ll= \leadsto
| l = \int \sinh a x \cosh a x \rd x
| r = \int \frac u a \rd ... | Primitive of Hyperbolic Sine of a x by Hyperbolic Cosine of a x/Proof 5 | https://proofwiki.org/wiki/Primitive_of_Hyperbolic_Sine_of_a_x | https://proofwiki.org/wiki/Primitive_of_Hyperbolic_Sine_of_a_x_by_Hyperbolic_Cosine_of_a_x/Proof_5 | [
"Primitives involving Hyperbolic Sine Function"
] | [] | [
"Derivative of Hyperbolic Sine of a x",
"Integration by Substitution",
"Primitive of Constant Multiple of Function",
"Primitive of Power"
] |
proofwiki-9761 | Primitive of Hyperbolic Cosine of a x | :$\ds \int \cosh a x \rd x = \frac {\sinh a x} a + C$ | {{begin-eqn}}
{{eqn | l = \int \cosh x \rd x
| r = \sinh x + C
| c = Primitive of $\cosh x$
}}
{{eqn | ll= \leadsto
| l = \int \cosh a x \rd x
| r = \frac 1 a \paren {\sinh a x} + C
| c = Primitive of Function of Constant Multiple
}}
{{eqn | r = \frac {\sinh a x} a + C
| c = simplify... | :$\ds \int \cosh a x \rd x = \frac {\sinh a x} a + C$ | {{begin-eqn}}
{{eqn | l = \int \cosh x \rd x
| r = \sinh x + C
| c = [[Primitive of Hyperbolic Cosine Function|Primitive of $\cosh x$]]
}}
{{eqn | ll= \leadsto
| l = \int \cosh a x \rd x
| r = \frac 1 a \paren {\sinh a x} + C
| c = [[Primitive of Function of Constant Multiple]]
}}
{{eqn | ... | Primitive of Hyperbolic Cosine of a x | https://proofwiki.org/wiki/Primitive_of_Hyperbolic_Cosine_of_a_x | https://proofwiki.org/wiki/Primitive_of_Hyperbolic_Cosine_of_a_x | [
"Primitives involving Hyperbolic Cosine Function"
] | [] | [
"Primitive of Hyperbolic Cosine Function",
"Primitive of Function of Constant Multiple"
] |
proofwiki-9762 | Primitive of Hyperbolic Tangent of a x | :$\ds \int \tanh a x \rd x = \frac {\map \ln {\cosh a x} } a + C$ | {{begin-eqn}}
{{eqn | l = \int \tanh x \rd x
| r = \map \ln {\cosh x} + C
| c = Primitive of $\tanh x$
}}
{{eqn | ll= \leadsto
| l = \int \tanh a x \rd x
| r = \frac 1 a \paren {\map \ln {\cosh a x} } + C
| c = Primitive of Function of Constant Multiple
}}
{{eqn | r = \frac {\map \ln {\cos... | :$\ds \int \tanh a x \rd x = \frac {\map \ln {\cosh a x} } a + C$ | {{begin-eqn}}
{{eqn | l = \int \tanh x \rd x
| r = \map \ln {\cosh x} + C
| c = [[Primitive of Hyperbolic Tangent Function|Primitive of $\tanh x$]]
}}
{{eqn | ll= \leadsto
| l = \int \tanh a x \rd x
| r = \frac 1 a \paren {\map \ln {\cosh a x} } + C
| c = [[Primitive of Function of Constan... | Primitive of Hyperbolic Tangent of a x | https://proofwiki.org/wiki/Primitive_of_Hyperbolic_Tangent_of_a_x | https://proofwiki.org/wiki/Primitive_of_Hyperbolic_Tangent_of_a_x | [
"Primitives involving Hyperbolic Tangent Function"
] | [] | [
"Primitive of Hyperbolic Tangent Function",
"Primitive of Function of Constant Multiple"
] |
proofwiki-9763 | Primitive of Hyperbolic Cotangent of a x | :$\ds \int \coth a x \rd x = \frac {\ln \size {\sinh a x} } a + C$ | {{begin-eqn}}
{{eqn | l = \int \coth x \rd x
| r = \ln \size {\sinh x} + C
| c = Primitive of $\coth x$
}}
{{eqn | ll= \leadsto
| l = \int \coth a x \rd x
| r = \frac 1 a \paren {\ln \size {\sinh a x} } + C
| c = Primitive of Function of Constant Multiple
}}
{{eqn | r = \frac {\ln \size {\... | :$\ds \int \coth a x \rd x = \frac {\ln \size {\sinh a x} } a + C$ | {{begin-eqn}}
{{eqn | l = \int \coth x \rd x
| r = \ln \size {\sinh x} + C
| c = [[Primitive of Hyperbolic Cotangent Function|Primitive of $\coth x$]]
}}
{{eqn | ll= \leadsto
| l = \int \coth a x \rd x
| r = \frac 1 a \paren {\ln \size {\sinh a x} } + C
| c = [[Primitive of Function of Con... | Primitive of Hyperbolic Cotangent of a x | https://proofwiki.org/wiki/Primitive_of_Hyperbolic_Cotangent_of_a_x | https://proofwiki.org/wiki/Primitive_of_Hyperbolic_Cotangent_of_a_x | [
"Primitives involving Hyperbolic Cotangent Function"
] | [] | [
"Primitive of Hyperbolic Cotangent Function",
"Primitive of Function of Constant Multiple"
] |
proofwiki-9764 | Primitive of Hyperbolic Cosecant of a x | :$\ds \int \csch a x \rd x = \frac 1 a \ln \size {\tanh \frac {a x} 2} + C$ | {{begin-eqn}}
{{eqn | l = \int \csch x \rd x
| r = \ln \size {\tanh \frac x 2} + C
| c = Primitive of $\csch x$
}}
{{eqn | ll= \leadsto
| l = \int \csch a x \rd x
| r = \frac 1 a \ln \size {\tanh \frac {a x} 2} + C
| c = Primitive of Function of Constant Multiple
}}
{{end-eqn}}
{{qed}} | :$\ds \int \csch a x \rd x = \frac 1 a \ln \size {\tanh \frac {a x} 2} + C$ | {{begin-eqn}}
{{eqn | l = \int \csch x \rd x
| r = \ln \size {\tanh \frac x 2} + C
| c = [[Primitive of Hyperbolic Cosecant Function/Hyperbolic Tangent Form|Primitive of $\csch x$]]
}}
{{eqn | ll= \leadsto
| l = \int \csch a x \rd x
| r = \frac 1 a \ln \size {\tanh \frac {a x} 2} + C
| c =... | Primitive of Hyperbolic Cosecant of a x | https://proofwiki.org/wiki/Primitive_of_Hyperbolic_Cosecant_of_a_x | https://proofwiki.org/wiki/Primitive_of_Hyperbolic_Cosecant_of_a_x | [
"Primitives involving Hyperbolic Cosecant Function"
] | [] | [
"Primitive of Hyperbolic Cosecant Function/Hyperbolic Tangent Form",
"Primitive of Function of Constant Multiple"
] |
proofwiki-9765 | Primitive of x by Hyperbolic Sine of a x | :$\ds \int x \sinh a x \rd x = \frac {x \cosh a x} a - \frac {\sinh a x} {a^2} + C$ | 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 = x
| c =
}}
{{eqn | ll= \leadsto
| l = \frac {\d u} {\d x}
| r = 1
| c = Derivative of Identity Function
}}
{{end-eqn... | :$\ds \int x \sinh a x \rd x = \frac {x \cosh a x} a - \frac {\sinh a x} {a^2} + C$ | 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 = x
| c =
}}
{{eqn | ll= \leadsto
| l = \frac {\d u} {\d x}
| r = 1
| c = [[Derivative of Id... | Primitive of x by Hyperbolic Sine of a x | https://proofwiki.org/wiki/Primitive_of_x_by_Hyperbolic_Sine_of_a_x | https://proofwiki.org/wiki/Primitive_of_x_by_Hyperbolic_Sine_of_a_x | [
"Primitives involving Hyperbolic Sine Function"
] | [] | [
"Definition:Primitive",
"Derivative of Identity Function",
"Primitive of Hyperbolic Sine of a x",
"Integration by Parts",
"Linear Combination of Integrals/Indefinite",
"Primitive of Hyperbolic Cosine of a x"
] |
proofwiki-9766 | Primitive of x by Hyperbolic Cosine of a x | :$\ds \int x \cosh a x \rd x = \frac {x \sinh a x} a - \frac {\cosh a x} {a^2} + C$ | With a view to expressing the primitive in the form:
:$\ds \int u \frac {\d v} {\d x} \rd x = u v - \int v \frac {\rd u} {\rd x} \rd x$
let:
{{begin-eqn}}
{{eqn | l = u
| r = x
| c =
}}
{{eqn | ll= \leadsto
| l = \frac {\d u} {\d x}
| r = 1
| c = Derivative of Identity Function
}}
{{end-e... | :$\ds \int x \cosh a x \rd x = \frac {x \sinh a x} a - \frac {\cosh a x} {a^2} + C$ | With a view to expressing the [[Definition:Primitive|primitive]] in the form:
:$\ds \int u \frac {\d v} {\d x} \rd x = u v - \int v \frac {\rd u} {\rd x} \rd x$
let:
{{begin-eqn}}
{{eqn | l = u
| r = x
| c =
}}
{{eqn | ll= \leadsto
| l = \frac {\d u} {\d x}
| r = 1
| c = [[Derivative of ... | Primitive of x by Hyperbolic Cosine of a x | https://proofwiki.org/wiki/Primitive_of_x_by_Hyperbolic_Cosine_of_a_x | https://proofwiki.org/wiki/Primitive_of_x_by_Hyperbolic_Cosine_of_a_x | [
"Primitives involving Hyperbolic Cosine Function"
] | [] | [
"Definition:Primitive",
"Derivative of Identity Function",
"Primitive of Hyperbolic Cosine of a x",
"Integration by Parts",
"Linear Combination of Integrals/Indefinite",
"Primitive of Hyperbolic Sine of a x"
] |
proofwiki-9767 | Primitive of x squared by Hyperbolic Sine of a x | :$\ds \int x^2 \sinh a x \rd x = \paren {\frac {x^2} a + \frac 2 {a^3} } \cosh a x - \frac {2 x \sinh a x} {a^2} + C$ | 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 = x^2
| c =
}}
{{eqn | ll= \leadsto
| l = \frac {\d u} {\d x}
| r = 2 x
| c = Derivative of Power
}}
{{end-eqn}}
and l... | :$\ds \int x^2 \sinh a x \rd x = \paren {\frac {x^2} a + \frac 2 {a^3} } \cosh a x - \frac {2 x \sinh a x} {a^2} + C$ | 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 = x^2
| c =
}}
{{eqn | ll= \leadsto
| l = \frac {\d u} {\d x}
| r = 2 x
| c = [[Derivative o... | Primitive of x squared by Hyperbolic Sine of a x | https://proofwiki.org/wiki/Primitive_of_x_squared_by_Hyperbolic_Sine_of_a_x | https://proofwiki.org/wiki/Primitive_of_x_squared_by_Hyperbolic_Sine_of_a_x | [
"Primitives involving Hyperbolic Sine Function"
] | [] | [
"Definition:Primitive",
"Power Rule for Derivatives",
"Primitive of Hyperbolic Sine of a x",
"Integration by Parts",
"Linear Combination of Integrals/Indefinite",
"Primitive of x by Hyperbolic Cosine of a x"
] |
proofwiki-9768 | Primitive of x squared by Hyperbolic Cosine of a x | :$\ds \int x^2 \cosh a x \rd x = \paren {\frac {x^2} a + \frac 2 {a^3} } \sinh a x - \frac {2 x \cosh a x} {a^2} + C$ | 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 = x^2
| c =
}}
{{eqn | ll= \leadsto
| l = \frac {\d u} {\d x}
| r = 2 x
| c = Derivative of Power
}}
{{end-eqn}}
and l... | :$\ds \int x^2 \cosh a x \rd x = \paren {\frac {x^2} a + \frac 2 {a^3} } \sinh a x - \frac {2 x \cosh a x} {a^2} + C$ | 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 = x^2
| c =
}}
{{eqn | ll= \leadsto
| l = \frac {\d u} {\d x}
| r = 2 x
| c = [[Derivative o... | Primitive of x squared by Hyperbolic Cosine of a x | https://proofwiki.org/wiki/Primitive_of_x_squared_by_Hyperbolic_Cosine_of_a_x | https://proofwiki.org/wiki/Primitive_of_x_squared_by_Hyperbolic_Cosine_of_a_x | [
"Primitives involving Hyperbolic Cosine Function"
] | [] | [
"Definition:Primitive",
"Power Rule for Derivatives",
"Primitive of Hyperbolic Cosine of a x",
"Integration by Parts",
"Linear Combination of Integrals/Indefinite",
"Primitive of x by Hyperbolic Cosine of a x"
] |
proofwiki-9769 | Primitive of Hyperbolic Tangent of a x over x | {{begin-eqn}}
{{eqn | l = \int \frac {\tanh a x \rd x} x
| r = \sum_{k \mathop \ge 1} \frac {2^{2 k} \paren {2^{2 k} - 1} B_{2 k} \paren {a x}^{2 k - 1} } {\paren {2 k - 1} \paren {2 k}!} + C
}}
{{eqn | r = a x - \frac {\paren {a x}^3} 9 + \frac {2 \paren {a x}^5} {75} - \cdots + C
}}
{{end-eqn}}
where $B_k$ deno... | {{begin-eqn}}
{{eqn | l = \tanh x
| r = \sum_{k \mathop = 1}^\infty \frac {2^{2 k} \paren {2^{2 k} - 1} B_{2 k} \, x^{2 k - 1} } {\paren {2 k}!}
| c = Power Series Expansion for Hyperbolic Tangent Function
}}
{{eqn | ll= \leadsto
| l = \frac {\tanh a x} x
| r = \sum_{k \mathop = 1}^\infty \frac ... | {{begin-eqn}}
{{eqn | l = \int \frac {\tanh a x \rd x} x
| r = \sum_{k \mathop \ge 1} \frac {2^{2 k} \paren {2^{2 k} - 1} B_{2 k} \paren {a x}^{2 k - 1} } {\paren {2 k - 1} \paren {2 k}!} + C
}}
{{eqn | r = a x - \frac {\paren {a x}^3} 9 + \frac {2 \paren {a x}^5} {75} - \cdots + C
}}
{{end-eqn}}
where $B_k$ deno... | {{begin-eqn}}
{{eqn | l = \tanh x
| r = \sum_{k \mathop = 1}^\infty \frac {2^{2 k} \paren {2^{2 k} - 1} B_{2 k} \, x^{2 k - 1} } {\paren {2 k}!}
| c = [[Power Series Expansion for Hyperbolic Tangent Function]]
}}
{{eqn | ll= \leadsto
| l = \frac {\tanh a x} x
| r = \sum_{k \mathop = 1}^\infty \f... | Primitive of Hyperbolic Tangent of a x over x | https://proofwiki.org/wiki/Primitive_of_Hyperbolic_Tangent_of_a_x_over_x | https://proofwiki.org/wiki/Primitive_of_Hyperbolic_Tangent_of_a_x_over_x | [
"Primitives involving Hyperbolic Tangent Function"
] | [
"Definition:Bernoulli Numbers"
] | [
"Power Series Expansion for Hyperbolic Tangent Function",
"Primitive of Power"
] |
proofwiki-9770 | Primitive of Hyperbolic Cotangent of a x over x | {{begin-eqn}}
{{eqn | l = \int \frac {\coth a x \rd x} x
| r = \sum_{k \mathop = 0}^\infty \frac {2^{2 k} B_{2 k} \paren {a x}^{2 k - 1} } {\paren {2 k}!} + C
}}
{{eqn | r = -\frac 1 {a x} + \frac {a x} 3 - \frac {\paren {a x}^3} {135} + \cdots + C
}}
{{end-eqn}}
where $B_k$ denotes the $k$th Bernoulli number. | {{begin-eqn}}
{{eqn | l = \coth x
| r = \sum_{k \mathop = 0}^\infty \frac {2^{2 k} B_{2 k} \, x^{2 k - 1} } {\paren {2 k}!}
| c = Power Series Expansion for Hyperbolic Cotangent Function
}}
{{eqn | r = \dfrac 1 x + \sum_{k \mathop = 1}^\infty \frac {2^{2 k} B_{2 k} \, x^{2 k - 1} } {\paren {2 k}!}
| c... | {{begin-eqn}}
{{eqn | l = \int \frac {\coth a x \rd x} x
| r = \sum_{k \mathop = 0}^\infty \frac {2^{2 k} B_{2 k} \paren {a x}^{2 k - 1} } {\paren {2 k}!} + C
}}
{{eqn | r = -\frac 1 {a x} + \frac {a x} 3 - \frac {\paren {a x}^3} {135} + \cdots + C
}}
{{end-eqn}}
where $B_k$ denotes the [[Definition:Bernoulli Num... | {{begin-eqn}}
{{eqn | l = \coth x
| r = \sum_{k \mathop = 0}^\infty \frac {2^{2 k} B_{2 k} \, x^{2 k - 1} } {\paren {2 k}!}
| c = [[Power Series Expansion for Hyperbolic Cotangent Function]]
}}
{{eqn | r = \dfrac 1 x + \sum_{k \mathop = 1}^\infty \frac {2^{2 k} B_{2 k} \, x^{2 k - 1} } {\paren {2 k}!}
... | Primitive of Hyperbolic Cotangent of a x over x | https://proofwiki.org/wiki/Primitive_of_Hyperbolic_Cotangent_of_a_x_over_x | https://proofwiki.org/wiki/Primitive_of_Hyperbolic_Cotangent_of_a_x_over_x | [
"Primitives involving Hyperbolic Cotangent Function"
] | [
"Definition:Bernoulli Numbers"
] | [
"Power Series Expansion for Hyperbolic Cotangent Function",
"Definition:Even Integer",
"Primitive of Power"
] |
proofwiki-9771 | Primitive of Hyperbolic Secant of a x over x | {{begin-eqn}}
{{eqn | l = \int \frac {\sech a x \rd x} x
| r = \ln \size x + \sum_{k \mathop \ge 1} \frac {E_{2 k} \paren {a x}^{2 k} } {\paren {2 k} \paren {2 k}!} + C
}}
{{eqn | r = \ln \size x - \frac {\paren {a x}^2} 4 + \frac {\paren {a x}^4} {96} - \frac {\paren {a x}^6} {4320} + \cdots + C
}}
{{end-eqn}}
w... | {{begin-eqn}}
{{eqn | l = \sech x
| r = \sum_{n \mathop = 0}^\infty \frac {E_{2 k} x^{2 k} } {\paren {2 k}!}
| c = Power Series Expansion for Hyperbolic Secant Function
}}
{{eqn | ll= \leadsto
| l = \frac {\sech a x} x
| r = \sum_{n \mathop = 0}^\infty \frac {E_{2 k} \paren {a x}^{2 k} } {x \par... | {{begin-eqn}}
{{eqn | l = \int \frac {\sech a x \rd x} x
| r = \ln \size x + \sum_{k \mathop \ge 1} \frac {E_{2 k} \paren {a x}^{2 k} } {\paren {2 k} \paren {2 k}!} + C
}}
{{eqn | r = \ln \size x - \frac {\paren {a x}^2} 4 + \frac {\paren {a x}^4} {96} - \frac {\paren {a x}^6} {4320} + \cdots + C
}}
{{end-eqn}}
w... | {{begin-eqn}}
{{eqn | l = \sech x
| r = \sum_{n \mathop = 0}^\infty \frac {E_{2 k} x^{2 k} } {\paren {2 k}!}
| c = [[Power Series Expansion for Hyperbolic Secant Function]]
}}
{{eqn | ll= \leadsto
| l = \frac {\sech a x} x
| r = \sum_{n \mathop = 0}^\infty \frac {E_{2 k} \paren {a x}^{2 k} } {x ... | Primitive of Hyperbolic Secant of a x over x | https://proofwiki.org/wiki/Primitive_of_Hyperbolic_Secant_of_a_x_over_x | https://proofwiki.org/wiki/Primitive_of_Hyperbolic_Secant_of_a_x_over_x | [
"Primitives involving Hyperbolic Secant Function"
] | [
"Definition:Euler Numbers"
] | [
"Power Series Expansion for Hyperbolic Secant Function",
"Primitive of Reciprocal",
"Primitive of Power"
] |
proofwiki-9772 | Primitive of Hyperbolic Cosecant of a x over x | {{begin-eqn}}
{{eqn | l = \int \frac {\csch a x \rd x} x
| r = \sum_{k \mathop = 0}^\infty \dfrac {2 \paren {1 - 2^{2 k - 1} } B_{2 k} \paren {a x}^{2 k - 1} } {\paren {2 k - 1}\paren {2 k}!} + C
}}
{{eqn | r = -\frac 1 {a x} - \frac {a x} 6 + \frac {7 \paren {a x}^3} {1080} + \cdots + C
}}
{{end-eqn}}
where $B_{... | {{begin-eqn}}
{{eqn | l = \csch x
| r = \sum_{k \mathop = 0}^\infty \dfrac {2 \paren {1 - 2^{2 k - 1} } B_{2 k} x^{2 k - 1} } {\paren {2 k}!}
| c = Power Series Expansion for Hyperbolic Cosecant Function
}}
{{eqn | ll= \leadsto
| l = \frac {\csch a x} x
| r = \sum_{k \mathop = 0}^\infty \dfrac {... | {{begin-eqn}}
{{eqn | l = \int \frac {\csch a x \rd x} x
| r = \sum_{k \mathop = 0}^\infty \dfrac {2 \paren {1 - 2^{2 k - 1} } B_{2 k} \paren {a x}^{2 k - 1} } {\paren {2 k - 1}\paren {2 k}!} + C
}}
{{eqn | r = -\frac 1 {a x} - \frac {a x} 6 + \frac {7 \paren {a x}^3} {1080} + \cdots + C
}}
{{end-eqn}}
where $B_{... | {{begin-eqn}}
{{eqn | l = \csch x
| r = \sum_{k \mathop = 0}^\infty \dfrac {2 \paren {1 - 2^{2 k - 1} } B_{2 k} x^{2 k - 1} } {\paren {2 k}!}
| c = [[Power Series Expansion for Hyperbolic Cosecant Function]]
}}
{{eqn | ll= \leadsto
| l = \frac {\csch a x} x
| r = \sum_{k \mathop = 0}^\infty \dfr... | Primitive of Hyperbolic Cosecant of a x over x | https://proofwiki.org/wiki/Primitive_of_Hyperbolic_Cosecant_of_a_x_over_x | https://proofwiki.org/wiki/Primitive_of_Hyperbolic_Cosecant_of_a_x_over_x | [
"Primitives involving Hyperbolic Cosecant Function"
] | [
"Definition:Bernoulli Numbers"
] | [
"Power Series Expansion for Hyperbolic Cosecant Function",
"Primitive of Power"
] |
proofwiki-9773 | Derivative of Hyperbolic Sine of a x | :$\map {D_x} {\sinh a x} = a \cosh a x$ | {{begin-eqn}}
{{eqn | l = \map {D_x} {\sinh x}
| r = \cosh x
| c = Derivative of $\sinh x$
}}
{{eqn | ll= \leadsto
| l = \map {D_x} {\sinh a x}
| r = a \cosh a x
| c = Derivative of Function of Constant Multiple
}}
{{end-eqn}}
{{qed}} | :$\map {D_x} {\sinh a x} = a \cosh a x$ | {{begin-eqn}}
{{eqn | l = \map {D_x} {\sinh x}
| r = \cosh x
| c = [[Derivative of Hyperbolic Sine|Derivative of $\sinh x$]]
}}
{{eqn | ll= \leadsto
| l = \map {D_x} {\sinh a x}
| r = a \cosh a x
| c = [[Derivative of Function of Constant Multiple]]
}}
{{end-eqn}}
{{qed}} | Derivative of Hyperbolic Sine of a x | https://proofwiki.org/wiki/Derivative_of_Hyperbolic_Sine_of_a_x | https://proofwiki.org/wiki/Derivative_of_Hyperbolic_Sine_of_a_x | [
"Derivatives of Hyperbolic Functions",
"Hyperbolic Sine Function"
] | [] | [
"Derivative of Hyperbolic Sine",
"Derivative of Function of Constant Multiple"
] |
proofwiki-9774 | Derivative of Hyperbolic Cosine of a x | :$\map {\dfrac \d {\d x} } {\cosh a x} = a \sinh a x$ | {{begin-eqn}}
{{eqn | l = \map {\dfrac \d {\d x} } {\cosh x}
| r = \sinh x
| c = Derivative of $\cosh x$
}}
{{eqn | ll= \leadsto
| l = \map {\dfrac \d {\d x} } {\cosh a x}
| r = a \sinh a x
| c = Derivative of Function of Constant Multiple
}}
{{end-eqn}}
{{qed}} | :$\map {\dfrac \d {\d x} } {\cosh a x} = a \sinh a x$ | {{begin-eqn}}
{{eqn | l = \map {\dfrac \d {\d x} } {\cosh x}
| r = \sinh x
| c = [[Derivative of Hyperbolic Cosine|Derivative of $\cosh x$]]
}}
{{eqn | ll= \leadsto
| l = \map {\dfrac \d {\d x} } {\cosh a x}
| r = a \sinh a x
| c = [[Derivative of Function of Constant Multiple]]
}}
{{end-e... | Derivative of Hyperbolic Cosine of a x | https://proofwiki.org/wiki/Derivative_of_Hyperbolic_Cosine_of_a_x | https://proofwiki.org/wiki/Derivative_of_Hyperbolic_Cosine_of_a_x | [
"Derivatives of Hyperbolic Functions",
"Hyperbolic Cosine Function"
] | [] | [
"Derivative of Hyperbolic Cosine",
"Derivative of Function of Constant Multiple"
] |
proofwiki-9775 | Primitive of Hyperbolic Sine of a x over x squared | :$\ds \int \frac {\sinh a x \ \d x} {x^2} = -\frac {\sinh a x} x + a \int \frac {\cosh a x \rd x} x$ | 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 = \sinh a x
| c =
}}
{{eqn | ll= \leadsto
| l = \frac {\d u} {\d x}
| r = a \cosh a x
| c = Derivative of $\sinh a x$
... | :$\ds \int \frac {\sinh a x \ \d x} {x^2} = -\frac {\sinh a x} x + a \int \frac {\cosh a x \rd x} x$ | 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 = \sinh a x
| c =
}}
{{eqn | ll= \leadsto
| l = \frac {\d u} {\d x}
| r = a \cosh a x
| c = ... | Primitive of Hyperbolic Sine of a x over x squared | https://proofwiki.org/wiki/Primitive_of_Hyperbolic_Sine_of_a_x_over_x_squared | https://proofwiki.org/wiki/Primitive_of_Hyperbolic_Sine_of_a_x_over_x_squared | [
"Primitives involving Hyperbolic Sine Function"
] | [] | [
"Definition:Primitive",
"Derivative of Hyperbolic Sine of a x",
"Primitive of Power",
"Integration by Parts"
] |
proofwiki-9776 | Primitive of Hyperbolic Cosine of a x over x squared | :$\ds \int \frac {\cosh a x \rd x} {x^2} = -\frac {\cosh a x} x + a \int \frac {\sinh a x \rd x} x$ | 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 = \cosh a x
| c =
}}
{{eqn | ll= \leadsto
| l = \frac {\d u} {\d x}
| r = a \sinh a x
| c = Derivative of $\cosh a x$
... | :$\ds \int \frac {\cosh a x \rd x} {x^2} = -\frac {\cosh a x} x + a \int \frac {\sinh a x \rd x} x$ | 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 = \cosh a x
| c =
}}
{{eqn | ll= \leadsto
| l = \frac {\d u} {\d x}
| r = a \sinh a x
| c = ... | Primitive of Hyperbolic Cosine of a x over x squared | https://proofwiki.org/wiki/Primitive_of_Hyperbolic_Cosine_of_a_x_over_x_squared | https://proofwiki.org/wiki/Primitive_of_Hyperbolic_Cosine_of_a_x_over_x_squared | [
"Primitives involving Hyperbolic Cosine Function"
] | [] | [
"Definition:Primitive",
"Derivative of Hyperbolic Cosine of a x",
"Primitive of Power",
"Integration by Parts"
] |
proofwiki-9777 | Primitive of Reciprocal of Hyperbolic Sine of a x | :$\ds \int \frac {\d x} {\sinh a x} = \frac 1 a \ln \size {\tanh \frac {a x} 2} + C$ | {{begin-eqn}}
{{eqn | l = \int \frac {\d x} {\sinh a x}
| r = \int \csch a x \rd x
| c = {{Defof|Hyperbolic Cosecant|index = 2}}
}}
{{eqn | r = \frac 1 a \ln \size {\tanh \frac {a x} 2} + C
| c = Primitive of $\csch a x$
}}
{{end-eqn}}
{{qed}} | :$\ds \int \frac {\d x} {\sinh a x} = \frac 1 a \ln \size {\tanh \frac {a x} 2} + C$ | {{begin-eqn}}
{{eqn | l = \int \frac {\d x} {\sinh a x}
| r = \int \csch a x \rd x
| c = {{Defof|Hyperbolic Cosecant|index = 2}}
}}
{{eqn | r = \frac 1 a \ln \size {\tanh \frac {a x} 2} + C
| c = [[Primitive of Hyperbolic Cosecant of a x|Primitive of $\csch a x$]]
}}
{{end-eqn}}
{{qed}} | Primitive of Reciprocal of Hyperbolic Sine of a x | https://proofwiki.org/wiki/Primitive_of_Reciprocal_of_Hyperbolic_Sine_of_a_x | https://proofwiki.org/wiki/Primitive_of_Reciprocal_of_Hyperbolic_Sine_of_a_x | [
"Primitives involving Hyperbolic Sine Function"
] | [] | [
"Primitive of Hyperbolic Cosecant of a x"
] |
proofwiki-9778 | Primitive of Reciprocal of Hyperbolic Cosine of a x | :$\ds \int \frac {\d x} {\cosh a x} = \frac {2 \map \arctan {e^{a x} } } a + C$ | {{begin-eqn}}
{{eqn | l = \int \frac {\d x} {\cosh a x}
| r = \int \sech a x \rd x
| c = {{Defof|Hyperbolic Secant|index = 2}}
}}
{{eqn | r = \frac {2 \map \arctan {e^{a x} } } a + C
| c = Primitive of $\sech a x$: Arctangent of Exponential Form
}}
{{end-eqn}}
{{qed}} | :$\ds \int \frac {\d x} {\cosh a x} = \frac {2 \map \arctan {e^{a x} } } a + C$ | {{begin-eqn}}
{{eqn | l = \int \frac {\d x} {\cosh a x}
| r = \int \sech a x \rd x
| c = {{Defof|Hyperbolic Secant|index = 2}}
}}
{{eqn | r = \frac {2 \map \arctan {e^{a x} } } a + C
| c = [[Primitive of Hyperbolic Secant of a x/Arctangent of Exponential Form|Primitive of $\sech a x$: Arctangent of Ex... | Primitive of Reciprocal of Hyperbolic Cosine of a x | https://proofwiki.org/wiki/Primitive_of_Reciprocal_of_Hyperbolic_Cosine_of_a_x | https://proofwiki.org/wiki/Primitive_of_Reciprocal_of_Hyperbolic_Cosine_of_a_x | [
"Primitives involving Hyperbolic Cosine Function"
] | [] | [
"Primitive of Hyperbolic Secant of a x/Arctangent of Exponential Form"
] |
proofwiki-9779 | Primitive of Reciprocal of Hyperbolic Tangent of a x | :$\ds \int \frac {\d x} {\tanh a x} = \frac {\ln \size {\sinh a x} } a + C$ | {{begin-eqn}}
{{eqn | l = \int \frac {\d x} {\tanh a x}
| r = \int \coth a x \rd x
| c = Hyperbolic Cotangent is Reciprocal of Hyperbolic Tangent
}}
{{eqn | r = \frac {\ln \size {\sinh a x} } a + C
| c = Primitive of $\coth a x$
}}
{{end-eqn}}
{{qed}} | :$\ds \int \frac {\d x} {\tanh a x} = \frac {\ln \size {\sinh a x} } a + C$ | {{begin-eqn}}
{{eqn | l = \int \frac {\d x} {\tanh a x}
| r = \int \coth a x \rd x
| c = [[Hyperbolic Cotangent is Reciprocal of Hyperbolic Tangent]]
}}
{{eqn | r = \frac {\ln \size {\sinh a x} } a + C
| c = [[Primitive of Hyperbolic Cotangent of a x|Primitive of $\coth a x$]]
}}
{{end-eqn}}
{{qed}} | Primitive of Reciprocal of Hyperbolic Tangent of a x | https://proofwiki.org/wiki/Primitive_of_Reciprocal_of_Hyperbolic_Tangent_of_a_x | https://proofwiki.org/wiki/Primitive_of_Reciprocal_of_Hyperbolic_Tangent_of_a_x | [
"Primitives involving Hyperbolic Tangent Function"
] | [] | [
"Hyperbolic Cotangent is Reciprocal of Hyperbolic Tangent",
"Primitive of Hyperbolic Cotangent of a x"
] |
proofwiki-9780 | Hyperbolic Cotangent is Reciprocal of Hyperbolic Tangent | :$\coth x = \dfrac 1 {\tanh x}$
where $\tanh$ and $\coth$ denote hyperbolic tangent and hyperbolic cotangent respectively. | {{begin-eqn}}
{{eqn | l = \coth x
| r = \frac {\cosh x} {\sinh x}
| c = {{Defof|Hyperbolic Cotangent|index = 2}}
}}
{{eqn | r = \frac 1 {\sinh x / \cosh x}
| c =
}}
{{eqn | r = \frac 1 {\tanh x}
| c = {{Defof|Hyperbolic Tangent|index = 2}}
}}
{{end-eqn}}
{{qed}} | :$\coth x = \dfrac 1 {\tanh x}$
where $\tanh$ and $\coth$ denote [[Definition:Hyperbolic Tangent|hyperbolic tangent]] and [[Definition:Hyperbolic Cotangent|hyperbolic cotangent]] respectively. | {{begin-eqn}}
{{eqn | l = \coth x
| r = \frac {\cosh x} {\sinh x}
| c = {{Defof|Hyperbolic Cotangent|index = 2}}
}}
{{eqn | r = \frac 1 {\sinh x / \cosh x}
| c =
}}
{{eqn | r = \frac 1 {\tanh x}
| c = {{Defof|Hyperbolic Tangent|index = 2}}
}}
{{end-eqn}}
{{qed}} | Hyperbolic Cotangent is Reciprocal of Hyperbolic Tangent | https://proofwiki.org/wiki/Hyperbolic_Cotangent_is_Reciprocal_of_Hyperbolic_Tangent | https://proofwiki.org/wiki/Hyperbolic_Cotangent_is_Reciprocal_of_Hyperbolic_Tangent | [
"Hyperbolic Cotangent Function",
"Hyperbolic Tangent Function"
] | [
"Definition:Hyperbolic Tangent",
"Definition:Hyperbolic Cotangent"
] | [] |
proofwiki-9781 | Primitive of Reciprocal of Hyperbolic Cotangent of a x | :$\ds \int \frac {\d x} {\coth a x} = \frac {\ln \size {\cosh a x} } a + C$ | {{begin-eqn}}
{{eqn | l = \int \frac {\d x} {\coth a x}
| r = \int \tanh a x \rd x
| c = Hyperbolic Cotangent is Reciprocal of Hyperbolic Tangent
}}
{{eqn | r = \frac {\ln \size {\cosh a x} } a + C
| c = Primitive of $\tanh a x$
}}
{{end-eqn}}
{{qed}} | :$\ds \int \frac {\d x} {\coth a x} = \frac {\ln \size {\cosh a x} } a + C$ | {{begin-eqn}}
{{eqn | l = \int \frac {\d x} {\coth a x}
| r = \int \tanh a x \rd x
| c = [[Hyperbolic Cotangent is Reciprocal of Hyperbolic Tangent]]
}}
{{eqn | r = \frac {\ln \size {\cosh a x} } a + C
| c = [[Primitive of Hyperbolic Tangent of a x|Primitive of $\tanh a x$]]
}}
{{end-eqn}}
{{qed}} | Primitive of Reciprocal of Hyperbolic Cotangent of a x | https://proofwiki.org/wiki/Primitive_of_Reciprocal_of_Hyperbolic_Cotangent_of_a_x | https://proofwiki.org/wiki/Primitive_of_Reciprocal_of_Hyperbolic_Cotangent_of_a_x | [
"Primitives involving Hyperbolic Cotangent Function"
] | [] | [
"Hyperbolic Cotangent is Reciprocal of Hyperbolic Tangent",
"Primitive of Hyperbolic Tangent of a x"
] |
proofwiki-9782 | Primitive of Reciprocal of Hyperbolic Secant of a x | :$\ds \int \frac {\d x} {\sech a x} = \frac {\sinh a x} a + C$ | {{begin-eqn}}
{{eqn | l = \int \frac {\d x} {\sech a x}
| r = \int \cosh a x \rd x
| c = {{Defof|Hyperbolic Secant|index = 2}}
}}
{{eqn | r = \frac {\sinh a x} a + C
| c = Primitive of $\cosh a x$
}}
{{end-eqn}}
{{qed}} | :$\ds \int \frac {\d x} {\sech a x} = \frac {\sinh a x} a + C$ | {{begin-eqn}}
{{eqn | l = \int \frac {\d x} {\sech a x}
| r = \int \cosh a x \rd x
| c = {{Defof|Hyperbolic Secant|index = 2}}
}}
{{eqn | r = \frac {\sinh a x} a + C
| c = [[Primitive of Hyperbolic Cosine of a x|Primitive of $\cosh a x$]]
}}
{{end-eqn}}
{{qed}} | Primitive of Reciprocal of Hyperbolic Secant of a x | https://proofwiki.org/wiki/Primitive_of_Reciprocal_of_Hyperbolic_Secant_of_a_x | https://proofwiki.org/wiki/Primitive_of_Reciprocal_of_Hyperbolic_Secant_of_a_x | [
"Primitives involving Hyperbolic Secant Function"
] | [] | [
"Primitive of Hyperbolic Cosine of a x"
] |
proofwiki-9783 | Primitive of Reciprocal of Hyperbolic Cosecant of a x | :$\ds \int \frac {\d x} {\csch a x} = \frac {\cosh a x} a + C$ | {{begin-eqn}}
{{eqn | l = \int \frac {\d x} {\csch a x}
| r = \int \sinh a x \rd x
| c = {{Defof|Hyperbolic Cosecant|index = 2}}
}}
{{eqn | r = \frac {\cosh a x} a + C
| c = Primitive of $\sinh a x$
}}
{{end-eqn}}
{{qed}} | :$\ds \int \frac {\d x} {\csch a x} = \frac {\cosh a x} a + C$ | {{begin-eqn}}
{{eqn | l = \int \frac {\d x} {\csch a x}
| r = \int \sinh a x \rd x
| c = {{Defof|Hyperbolic Cosecant|index = 2}}
}}
{{eqn | r = \frac {\cosh a x} a + C
| c = [[Primitive of Hyperbolic Sine of a x|Primitive of $\sinh a x$]]
}}
{{end-eqn}}
{{qed}} | Primitive of Reciprocal of Hyperbolic Cosecant of a x | https://proofwiki.org/wiki/Primitive_of_Reciprocal_of_Hyperbolic_Cosecant_of_a_x | https://proofwiki.org/wiki/Primitive_of_Reciprocal_of_Hyperbolic_Cosecant_of_a_x | [
"Primitives involving Hyperbolic Cosecant Function"
] | [] | [
"Primitive of Hyperbolic Sine of a x"
] |
proofwiki-9784 | Primitive of x over Hyperbolic Sine of a x | {{begin-eqn}}
{{eqn | l = \int \frac {x \rd x} {\sinh a x}
| r = \frac 1 {a^2} \sum_{n \mathop = 0}^\infty \frac {2 \paren {1 - 2^{2 n - 1} } B_{2 n} \paren {a x}^{2 n + 1} } {\paren {2 n + 1}!} + C
}}
{{eqn | r = \dfrac 1 {a^2} \paren {a x - \dfrac {\paren {a x}^3} {18} + \dfrac {7 \paren {a x}^5} {1800} - \cdot... | {{begin-eqn}}
{{eqn | l = \int \frac {x \rd x} {\sinh a x}
| r = \int x \csch a x \rd x
| c = {{Defof|Hyperbolic Cosecant|index = 2}}
}}
{{eqn | r = \frac 1 {a^2} \sum_{n \mathop = 0}^\infty \frac {2 \paren {1 - 2^{2 n - 1} } B_{2 n} \paren {a x}^{2 n + 1} } {\paren {2 n + 1}!} + C
| c = Primitive of ... | {{begin-eqn}}
{{eqn | l = \int \frac {x \rd x} {\sinh a x}
| r = \frac 1 {a^2} \sum_{n \mathop = 0}^\infty \frac {2 \paren {1 - 2^{2 n - 1} } B_{2 n} \paren {a x}^{2 n + 1} } {\paren {2 n + 1}!} + C
}}
{{eqn | r = \dfrac 1 {a^2} \paren {a x - \dfrac {\paren {a x}^3} {18} + \dfrac {7 \paren {a x}^5} {1800} - \cdot... | {{begin-eqn}}
{{eqn | l = \int \frac {x \rd x} {\sinh a x}
| r = \int x \csch a x \rd x
| c = {{Defof|Hyperbolic Cosecant|index = 2}}
}}
{{eqn | r = \frac 1 {a^2} \sum_{n \mathop = 0}^\infty \frac {2 \paren {1 - 2^{2 n - 1} } B_{2 n} \paren {a x}^{2 n + 1} } {\paren {2 n + 1}!} + C
| c = [[Primitive o... | Primitive of x over Hyperbolic Sine of a x | https://proofwiki.org/wiki/Primitive_of_x_over_Hyperbolic_Sine_of_a_x | https://proofwiki.org/wiki/Primitive_of_x_over_Hyperbolic_Sine_of_a_x | [
"Primitives involving Hyperbolic Sine Function"
] | [
"Definition:Bernoulli Numbers"
] | [
"Primitive of x by Hyperbolic Cosecant of a x"
] |
proofwiki-9785 | Primitive of x over Hyperbolic Cosine of a x | {{begin-eqn}}
{{eqn | l = \int \frac {x \rd x} {\cosh a x}
| r = \frac 1 {a^2} \sum_{n \mathop = 0}^\infty \frac {E_{2 n} \paren {a x}^{2 n + 2} } {\paren {2 n + 2} \paren {2 n}!} + C
}}
{{eqn | r = \frac 1 {a^2} \paren {\frac {\paren {a x}^2} 2 - \frac {\paren {a x}^4} 8 + \frac {5 \paren {a x}^6} {144} - \cdots... | {{begin-eqn}}
{{eqn | l = \int \frac {x \rd x} {\cosh a x}
| r = \int x \sech a x \rd x
| c = {{Defof|Hyperbolic Secant|index = 2}}
}}
{{eqn | r = \frac 1 {a^2} \sum_{n \mathop = 0}^\infty \frac {E_{2 n} \paren {a x}^{2 n + 2} } {\paren {2 n + 2} \paren {2 n}!} + C
| c = Primitive of $x \sech a x$
}}
... | {{begin-eqn}}
{{eqn | l = \int \frac {x \rd x} {\cosh a x}
| r = \frac 1 {a^2} \sum_{n \mathop = 0}^\infty \frac {E_{2 n} \paren {a x}^{2 n + 2} } {\paren {2 n + 2} \paren {2 n}!} + C
}}
{{eqn | r = \frac 1 {a^2} \paren {\frac {\paren {a x}^2} 2 - \frac {\paren {a x}^4} 8 + \frac {5 \paren {a x}^6} {144} - \cdots... | {{begin-eqn}}
{{eqn | l = \int \frac {x \rd x} {\cosh a x}
| r = \int x \sech a x \rd x
| c = {{Defof|Hyperbolic Secant|index = 2}}
}}
{{eqn | r = \frac 1 {a^2} \sum_{n \mathop = 0}^\infty \frac {E_{2 n} \paren {a x}^{2 n + 2} } {\paren {2 n + 2} \paren {2 n}!} + C
| c = [[Primitive of x by Hyperbolic... | Primitive of x over Hyperbolic Cosine of a x | https://proofwiki.org/wiki/Primitive_of_x_over_Hyperbolic_Cosine_of_a_x | https://proofwiki.org/wiki/Primitive_of_x_over_Hyperbolic_Cosine_of_a_x | [
"Primitives involving Hyperbolic Cosine Function"
] | [
"Definition:Euler Numbers"
] | [
"Primitive of x by Hyperbolic Secant of a x"
] |
proofwiki-9786 | Primitive of Square of Hyperbolic Sine of a x | :$\ds \int \sinh^2 a x \rd x = \dfrac {\sinh a x \cosh a x} {2 a} - \frac x 2 + C$ | {{begin-eqn}}
{{eqn | l = \int \sinh^2 x \rd x
| r = \frac {\sinh x \cosh x - x} 2 + C
| c = Primitive of $\sinh^2 x$
}}
{{eqn | ll= \leadsto
| l = \int \sinh^2 a x \rd x
| r = \frac 1 a \paren {\frac {\sinh a x \cosh a x - a x} 2} + C
| c = Primitive of Function of Constant Multiple
}}
{{... | :$\ds \int \sinh^2 a x \rd x = \dfrac {\sinh a x \cosh a x} {2 a} - \frac x 2 + C$ | {{begin-eqn}}
{{eqn | l = \int \sinh^2 x \rd x
| r = \frac {\sinh x \cosh x - x} 2 + C
| c = [[Primitive of Square of Hyperbolic Sine Function/Corollary|Primitive of $\sinh^2 x$]]
}}
{{eqn | ll= \leadsto
| l = \int \sinh^2 a x \rd x
| r = \frac 1 a \paren {\frac {\sinh a x \cosh a x - a x} 2} + ... | Primitive of Square of Hyperbolic Sine of a x | https://proofwiki.org/wiki/Primitive_of_Square_of_Hyperbolic_Sine_of_a_x | https://proofwiki.org/wiki/Primitive_of_Square_of_Hyperbolic_Sine_of_a_x | [
"Primitive of Square of Hyperbolic Sine of a x",
"Primitives involving Hyperbolic Sine Function"
] | [] | [
"Primitive of Square of Hyperbolic Sine Function/Corollary",
"Primitive of Function of Constant Multiple"
] |
proofwiki-9787 | Primitive of Square of Hyperbolic Cosine of a x | :$\ds \int \cosh^2 a x \rd x = \frac {\sinh a x \cosh a x} {2 a} + \frac x 2 + C$ | {{begin-eqn}}
{{eqn | l = \int \cosh^2 x \rd x
| r = \frac {\sinh x \cosh x + x} 2 + C
| c = Primitive of $\cosh^2 x$
}}
{{eqn | ll= \leadsto
| l = \int \cosh^2 a x \rd x
| r = \frac 1 a \paren {\frac {\sinh a x \cosh a x + a x} 2} + C
| c = Primitive of Function of Constant Multiple
}}
{{... | :$\ds \int \cosh^2 a x \rd x = \frac {\sinh a x \cosh a x} {2 a} + \frac x 2 + C$ | {{begin-eqn}}
{{eqn | l = \int \cosh^2 x \rd x
| r = \frac {\sinh x \cosh x + x} 2 + C
| c = [[Primitive of Square of Hyperbolic Cosine Function/Corollary|Primitive of $\cosh^2 x$]]
}}
{{eqn | ll= \leadsto
| l = \int \cosh^2 a x \rd x
| r = \frac 1 a \paren {\frac {\sinh a x \cosh a x + a x} 2} ... | Primitive of Square of Hyperbolic Cosine of a x | https://proofwiki.org/wiki/Primitive_of_Square_of_Hyperbolic_Cosine_of_a_x | https://proofwiki.org/wiki/Primitive_of_Square_of_Hyperbolic_Cosine_of_a_x | [
"Primitive of Square of Hyperbolic Cosine of a x",
"Primitives involving Hyperbolic Cosine Function"
] | [] | [
"Primitive of Square of Hyperbolic Cosine Function/Corollary",
"Primitive of Function of Constant Multiple"
] |
proofwiki-9788 | Primitive of Square of Hyperbolic Tangent of a x | :$\ds \int \tanh^2 a x \rd x = x - \frac {\tanh a x} a + C$ | {{begin-eqn}}
{{eqn | l = \int \tanh^2 x \rd x
| r = x - \tanh x + C
| c = Primitive of $\tanh^2 x$
}}
{{eqn | ll= \leadsto
| l = \int \tanh^2 a x \rd x
| r = \frac 1 a \paren {a x - \tanh a x} + C
| c = Primitive of Function of Constant Multiple
}}
{{eqn | r = x - \frac {\tanh a x} a + C
... | :$\ds \int \tanh^2 a x \rd x = x - \frac {\tanh a x} a + C$ | {{begin-eqn}}
{{eqn | l = \int \tanh^2 x \rd x
| r = x - \tanh x + C
| c = [[Primitive of Square of Hyperbolic Tangent Function|Primitive of $\tanh^2 x$]]
}}
{{eqn | ll= \leadsto
| l = \int \tanh^2 a x \rd x
| r = \frac 1 a \paren {a x - \tanh a x} + C
| c = [[Primitive of Function of Cons... | Primitive of Square of Hyperbolic Tangent of a x | https://proofwiki.org/wiki/Primitive_of_Square_of_Hyperbolic_Tangent_of_a_x | https://proofwiki.org/wiki/Primitive_of_Square_of_Hyperbolic_Tangent_of_a_x | [
"Primitives involving Hyperbolic Tangent Function"
] | [] | [
"Primitive of Square of Hyperbolic Tangent Function",
"Primitive of Function of Constant Multiple"
] |
proofwiki-9789 | Primitive of Square of Hyperbolic Cotangent of a x | :$\ds \int \coth^2 a x \rd x = x - \frac {\coth a x} a + C$ | {{begin-eqn}}
{{eqn | l = \int \coth^2 x \rd x
| r = x - \coth x + C
| c = Primitive of $\coth^2 x$
}}
{{eqn | ll= \leadsto
| l = \int \coth^2 a x \rd x
| r = \frac 1 a \paren {a x - \coth a x} + C
| c = Primitive of Function of Constant Multiple
}}
{{eqn | r = x - \frac {\coth a x} a + C
... | :$\ds \int \coth^2 a x \rd x = x - \frac {\coth a x} a + C$ | {{begin-eqn}}
{{eqn | l = \int \coth^2 x \rd x
| r = x - \coth x + C
| c = [[Primitive of Square of Hyperbolic Cotangent Function|Primitive of $\coth^2 x$]]
}}
{{eqn | ll= \leadsto
| l = \int \coth^2 a x \rd x
| r = \frac 1 a \paren {a x - \coth a x} + C
| c = [[Primitive of Function of Co... | Primitive of Square of Hyperbolic Cotangent of a x | https://proofwiki.org/wiki/Primitive_of_Square_of_Hyperbolic_Cotangent_of_a_x | https://proofwiki.org/wiki/Primitive_of_Square_of_Hyperbolic_Cotangent_of_a_x | [
"Primitives involving Hyperbolic Cotangent Function"
] | [] | [
"Primitive of Square of Hyperbolic Cotangent Function",
"Primitive of Function of Constant Multiple"
] |
proofwiki-9790 | Primitive of Square of Hyperbolic Secant of a x | :$\ds \int \sech^2 a x \rd x = \frac {\tanh a x} a + C$ | {{begin-eqn}}
{{eqn | l = \int \sech^2 x \rd x
| r = \tanh x + C
| c = Primitive of $\sech^2 x$
}}
{{eqn | ll= \leadsto
| l = \int \sech^2 a x \rd x
| r = \frac 1 a \paren {\tanh a x} + C
| c = Primitive of Function of Constant Multiple
}}
{{eqn | r = \frac {\tanh a x} a + C
| c = si... | :$\ds \int \sech^2 a x \rd x = \frac {\tanh a x} a + C$ | {{begin-eqn}}
{{eqn | l = \int \sech^2 x \rd x
| r = \tanh x + C
| c = [[Primitive of Square of Hyperbolic Secant Function|Primitive of $\sech^2 x$]]
}}
{{eqn | ll= \leadsto
| l = \int \sech^2 a x \rd x
| r = \frac 1 a \paren {\tanh a x} + C
| c = [[Primitive of Function of Constant Multip... | Primitive of Square of Hyperbolic Secant of a x | https://proofwiki.org/wiki/Primitive_of_Square_of_Hyperbolic_Secant_of_a_x | https://proofwiki.org/wiki/Primitive_of_Square_of_Hyperbolic_Secant_of_a_x | [
"Primitives involving Hyperbolic Secant Function"
] | [] | [
"Primitive of Square of Hyperbolic Secant Function",
"Primitive of Function of Constant Multiple"
] |
proofwiki-9791 | Primitive of Square of Hyperbolic Cosecant of a x | :$\ds \int \csch^2 a x \rd x = \frac {-\coth a x} a + C$ | {{begin-eqn}}
{{eqn | l = \int \csch^2 x \rd x
| r = -\coth x + C
| c = Primitive of $\csch^2 x$
}}
{{eqn | ll= \leadsto
| l = \int \csch^2 a x \rd x
| r = \frac 1 a \paren {-\coth a x} + C
| c = Primitive of Function of Constant Multiple
}}
{{eqn | r = \frac {-\coth a x} a + C
| c =... | :$\ds \int \csch^2 a x \rd x = \frac {-\coth a x} a + C$ | {{begin-eqn}}
{{eqn | l = \int \csch^2 x \rd x
| r = -\coth x + C
| c = [[Primitive of Square of Hyperbolic Cosecant Function|Primitive of $\csch^2 x$]]
}}
{{eqn | ll= \leadsto
| l = \int \csch^2 a x \rd x
| r = \frac 1 a \paren {-\coth a x} + C
| c = [[Primitive of Function of Constant Mu... | Primitive of Square of Hyperbolic Cosecant of a x | https://proofwiki.org/wiki/Primitive_of_Square_of_Hyperbolic_Cosecant_of_a_x | https://proofwiki.org/wiki/Primitive_of_Square_of_Hyperbolic_Cosecant_of_a_x | [
"Primitives involving Hyperbolic Cosecant Function"
] | [] | [
"Primitive of Square of Hyperbolic Cosecant Function",
"Primitive of Function of Constant Multiple"
] |
proofwiki-9792 | Primitive of Square of Hyperbolic Sine of a x/Corollary | :$\ds \int \sinh^2 a x \rd x = \frac {\sinh 2 a x} {4 a} - \frac x 2 + C$ | {{begin-eqn}}
{{eqn | l = \int \sinh^2 a x \rd x
| r = \frac {\sinh a x \cosh a x} {2 a} - \frac x 2 + C
| c = Primitive of $\sinh^2 a x$
}}
{{eqn | r = \frac {\frac {\sinh 2 a x} 2} {2 a} - \frac x 2 + C
| c = Double Angle Formula for Hyperbolic Sine
}}
{{eqn | r = \dfrac {\sinh 2 a x} {4 a} - \frac ... | :$\ds \int \sinh^2 a x \rd x = \frac {\sinh 2 a x} {4 a} - \frac x 2 + C$ | {{begin-eqn}}
{{eqn | l = \int \sinh^2 a x \rd x
| r = \frac {\sinh a x \cosh a x} {2 a} - \frac x 2 + C
| c = [[Primitive of Square of Hyperbolic Sine of a x|Primitive of $\sinh^2 a x$]]
}}
{{eqn | r = \frac {\frac {\sinh 2 a x} 2} {2 a} - \frac x 2 + C
| c = [[Double Angle Formula for Hyperbolic Sin... | Primitive of Square of Hyperbolic Sine of a x/Corollary | https://proofwiki.org/wiki/Primitive_of_Square_of_Hyperbolic_Sine_of_a_x/Corollary | https://proofwiki.org/wiki/Primitive_of_Square_of_Hyperbolic_Sine_of_a_x/Corollary | [
"Primitive of Square of Hyperbolic Sine of a x"
] | [] | [
"Primitive of Square of Hyperbolic Sine of a x",
"Double Angle Formulas/Hyperbolic Sine"
] |
proofwiki-9793 | Primitive of x by Square of Hyperbolic Sine of a x | :$\ds \int x \sinh^2 a x \rd x = \dfrac {x \sinh 2 a x} {4 a} - \frac {\cosh 2 a x} {8 a^2} - \frac {x^2} 4 + C$ | 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 = x
| c =
}}
{{eqn | ll= \leadsto
| l = \frac {\d u} {\d x}
| r = 1
| c = Derivative of Identity Function
}}
{{end-eqn... | :$\ds \int x \sinh^2 a x \rd x = \dfrac {x \sinh 2 a x} {4 a} - \frac {\cosh 2 a x} {8 a^2} - \frac {x^2} 4 + C$ | 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 = x
| c =
}}
{{eqn | ll= \leadsto
| l = \frac {\d u} {\d x}
| r = 1
| c = [[Derivative of Id... | Primitive of x by Square of Hyperbolic Sine of a x | https://proofwiki.org/wiki/Primitive_of_x_by_Square_of_Hyperbolic_Sine_of_a_x | https://proofwiki.org/wiki/Primitive_of_x_by_Square_of_Hyperbolic_Sine_of_a_x | [
"Primitives involving Hyperbolic Sine Function"
] | [] | [
"Definition:Primitive",
"Derivative of Identity Function",
"Primitive of Square of Hyperbolic Sine of a x/Corollary",
"Integration by Parts",
"Linear Combination of Integrals/Indefinite",
"Primitive of Power",
"Primitive of Hyperbolic Sine of a x"
] |
proofwiki-9794 | Primitive of Square of Hyperbolic Cosine of a x/Corollary | :$\ds \int \cosh^2 a x \rd x = \frac {\sinh 2 a x} {4 a} + \frac x 2 + C$ | {{begin-eqn}}
{{eqn | l = \int \cosh^2 a x \rd x
| r = \frac {\sinh a x \cosh a x} {2 a} + \frac x 2 + C
| c = Primitive of $\cosh^2 a x$
}}
{{eqn | r = \frac {\frac {\sinh 2 a x} 2} {2 a} + \frac x 2 + C
| c = Double Angle Formula for Hyperbolic Sine
}}
{{eqn | r = \dfrac {\sinh 2 a x} {4 a} + \frac ... | :$\ds \int \cosh^2 a x \rd x = \frac {\sinh 2 a x} {4 a} + \frac x 2 + C$ | {{begin-eqn}}
{{eqn | l = \int \cosh^2 a x \rd x
| r = \frac {\sinh a x \cosh a x} {2 a} + \frac x 2 + C
| c = [[Primitive of Square of Hyperbolic Cosine of a x|Primitive of $\cosh^2 a x$]]
}}
{{eqn | r = \frac {\frac {\sinh 2 a x} 2} {2 a} + \frac x 2 + C
| c = [[Double Angle Formula for Hyperbolic S... | Primitive of Square of Hyperbolic Cosine of a x/Corollary | https://proofwiki.org/wiki/Primitive_of_Square_of_Hyperbolic_Cosine_of_a_x/Corollary | https://proofwiki.org/wiki/Primitive_of_Square_of_Hyperbolic_Cosine_of_a_x/Corollary | [
"Primitive of Square of Hyperbolic Cosine of a x"
] | [] | [
"Primitive of Square of Hyperbolic Cosine of a x",
"Double Angle Formulas/Hyperbolic Sine"
] |
proofwiki-9795 | Primitive of x by Square of Hyperbolic Cosine of a x | :$\ds \int x \cosh^2 a x \rd x = \frac {x \sinh 2 a x} {4 a} - \frac {\cosh 2 a x} {8 a^2} + \frac {x^2} 4 + C$ | 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 = x
| c =
}}
{{eqn | ll= \leadsto
| l = \frac {\d u} {\d x}
| r = 1
| c = Derivative of Identity Function
}}
{{end-eqn... | :$\ds \int x \cosh^2 a x \rd x = \frac {x \sinh 2 a x} {4 a} - \frac {\cosh 2 a x} {8 a^2} + \frac {x^2} 4 + C$ | 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 = x
| c =
}}
{{eqn | ll= \leadsto
| l = \frac {\d u} {\d x}
| r = 1
| c = [[Derivative of Id... | Primitive of x by Square of Hyperbolic Cosine of a x | https://proofwiki.org/wiki/Primitive_of_x_by_Square_of_Hyperbolic_Cosine_of_a_x | https://proofwiki.org/wiki/Primitive_of_x_by_Square_of_Hyperbolic_Cosine_of_a_x | [
"Primitives involving Hyperbolic Cosine Function"
] | [] | [
"Definition:Primitive",
"Derivative of Identity Function",
"Primitive of Square of Hyperbolic Cosine of a x/Corollary",
"Integration by Parts",
"Linear Combination of Integrals/Indefinite",
"Primitive of Power",
"Primitive of Hyperbolic Sine of a x"
] |
proofwiki-9796 | Primitive of x by Square of Hyperbolic Tangent of a x | :$\ds \int x \tanh^2 a x \rd x = \frac {x^2} 2 - \frac {x \tanh a x} a + \frac 1 {a^2} \ln \size {\cosh a x} + C$ | 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 = x
| c =
}}
{{eqn | ll= \leadsto
| l = \frac {\d u} {\d x}
| r = 1
| c = Derivative of Identity Function
}}
{{end-eqn... | :$\ds \int x \tanh^2 a x \rd x = \frac {x^2} 2 - \frac {x \tanh a x} a + \frac 1 {a^2} \ln \size {\cosh a x} + C$ | 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 = x
| c =
}}
{{eqn | ll= \leadsto
| l = \frac {\d u} {\d x}
| r = 1
| c = [[Derivative of Id... | Primitive of x by Square of Hyperbolic Tangent of a x | https://proofwiki.org/wiki/Primitive_of_x_by_Square_of_Hyperbolic_Tangent_of_a_x | https://proofwiki.org/wiki/Primitive_of_x_by_Square_of_Hyperbolic_Tangent_of_a_x | [
"Primitives involving Hyperbolic Tangent Function"
] | [] | [
"Definition:Primitive",
"Derivative of Identity Function",
"Primitive of Square of Hyperbolic Tangent of a x",
"Integration by Parts",
"Linear Combination of Integrals/Indefinite",
"Primitive of Power",
"Primitive of Hyperbolic Tangent of a x"
] |
proofwiki-9797 | Primitive of x by Square of Hyperbolic Cotangent of a x | :$\ds \int x \coth^2 a x \rd x = \frac {x^2} 2 - \frac {x \coth a x} a + \frac 1 {a^2} \ln \size {\sinh a x} + C$ | 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 = x
| c =
}}
{{eqn | ll= \leadsto
| l = \frac {\d u} {\d x}
| r = 1
| c = Derivative of Identity Function
}}
{{end-eqn... | :$\ds \int x \coth^2 a x \rd x = \frac {x^2} 2 - \frac {x \coth a x} a + \frac 1 {a^2} \ln \size {\sinh a x} + C$ | 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 = x
| c =
}}
{{eqn | ll= \leadsto
| l = \frac {\d u} {\d x}
| r = 1
| c = [[Derivative of Id... | Primitive of x by Square of Hyperbolic Cotangent of a x | https://proofwiki.org/wiki/Primitive_of_x_by_Square_of_Hyperbolic_Cotangent_of_a_x | https://proofwiki.org/wiki/Primitive_of_x_by_Square_of_Hyperbolic_Cotangent_of_a_x | [
"Primitives involving Hyperbolic Cotangent Function"
] | [] | [
"Definition:Primitive",
"Derivative of Identity Function",
"Primitive of Square of Hyperbolic Cotangent of a x",
"Integration by Parts",
"Linear Combination of Integrals/Indefinite",
"Primitive of Power",
"Primitive of Hyperbolic Cotangent of a x"
] |
proofwiki-9798 | Primitive of x by Square of Hyperbolic Secant of a x | :$\ds \int x \sech^2 a x \rd x = \frac {x \tanh a x} a - \frac 1 {a^2} \ln \size {\cosh a x} + C$ | 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 = x
| c =
}}
{{eqn | ll= \leadsto
| l = \frac {\d u} {\d x}
| r = 1
| c = Derivative of Identity Function
}}
{{end-eqn... | :$\ds \int x \sech^2 a x \rd x = \frac {x \tanh a x} a - \frac 1 {a^2} \ln \size {\cosh a x} + C$ | 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 = x
| c =
}}
{{eqn | ll= \leadsto
| l = \frac {\d u} {\d x}
| r = 1
| c = [[Derivative of Id... | Primitive of x by Square of Hyperbolic Secant of a x | https://proofwiki.org/wiki/Primitive_of_x_by_Square_of_Hyperbolic_Secant_of_a_x | https://proofwiki.org/wiki/Primitive_of_x_by_Square_of_Hyperbolic_Secant_of_a_x | [
"Primitives involving Hyperbolic Secant Function"
] | [] | [
"Definition:Primitive",
"Derivative of Identity Function",
"Primitive of Square of Hyperbolic Secant of a x",
"Integration by Parts",
"Linear Combination of Integrals/Indefinite",
"Primitive of Hyperbolic Tangent of a x"
] |
proofwiki-9799 | Primitive of x by Square of Hyperbolic Cosecant of a x | :$\ds \int x \csch^2 a x \rd x = \frac {-x \coth a x} a + \frac 1 {a^2} \ln \size {\sinh a x} + C$ | 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 = x
| c =
}}
{{eqn | ll= \leadsto
| l = \frac {\d u} {\d x}
| r = 1
| c = Derivative of Identity Function
}}
{{end-eqn... | :$\ds \int x \csch^2 a x \rd x = \frac {-x \coth a x} a + \frac 1 {a^2} \ln \size {\sinh a x} + C$ | 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 = x
| c =
}}
{{eqn | ll= \leadsto
| l = \frac {\d u} {\d x}
| r = 1
| c = [[Derivative of Id... | Primitive of x by Square of Hyperbolic Cosecant of a x | https://proofwiki.org/wiki/Primitive_of_x_by_Square_of_Hyperbolic_Cosecant_of_a_x | https://proofwiki.org/wiki/Primitive_of_x_by_Square_of_Hyperbolic_Cosecant_of_a_x | [
"Primitives involving Hyperbolic Cosecant Function"
] | [] | [
"Definition:Primitive",
"Derivative of Identity Function",
"Primitive of Square of Hyperbolic Cosecant of a x",
"Integration by Parts",
"Linear Combination of Integrals/Indefinite",
"Primitive of Hyperbolic Cotangent of a x"
] |
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