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-9100 | Primitive of Function of Root of a x + b | :$\ds \int \map F {\sqrt {a x + b} } \rd x = \frac 2 a \int u \map F u \rd u$
where $u = \sqrt {a x + b}$. | {{begin-eqn}}
{{eqn | l = u
| r = \sqrt {a x + b}
| c =
}}
{{eqn | l = u
| r = \paren {a x + b}^{1/2}
| c =
}}
{{eqn | ll= \leadsto
| l = \frac {\d u} {\d x}
| r = \frac 1 2 \paren {a x + b}^{-1/2} \map {\frac \d {\d x} } {a x + b}
| c = Chain Rule for Derivatives, Power Ru... | :$\ds \int \map F {\sqrt {a x + b} } \rd x = \frac 2 a \int u \map F u \rd u$
where $u = \sqrt {a x + b}$. | {{begin-eqn}}
{{eqn | l = u
| r = \sqrt {a x + b}
| c =
}}
{{eqn | l = u
| r = \paren {a x + b}^{1/2}
| c =
}}
{{eqn | ll= \leadsto
| l = \frac {\d u} {\d x}
| r = \frac 1 2 \paren {a x + b}^{-1/2} \map {\frac \d {\d x} } {a x + b}
| c = [[Chain Rule for Derivatives]], [[Po... | Primitive of Function of Root of a x + b | https://proofwiki.org/wiki/Primitive_of_Function_of_Root_of_a_x_+_b | https://proofwiki.org/wiki/Primitive_of_Function_of_Root_of_a_x_+_b | [
"Integral Substitutions",
"Primitives involving Root of a x + b"
] | [] | [
"Derivative of Composite Function",
"Power Rule for Derivatives",
"Primitive of Composite Function",
"Primitive of Constant Multiple of Function"
] |
proofwiki-9101 | Derivative of Nth Root | Let $n \in \N_{>0}$.
Let $f: \R \to \R$ be the real function defined as $\map f x = \sqrt [n] x$.
Then:
:$\map {f'} x = \dfrac 1 {n \paren {\sqrt [n] x}^{n - 1} }$
everywhere that $\map f x = \sqrt [n] x$ is defined. | {{begin-eqn}}
{{eqn | l = \map f x
| r = \sqrt [n] x
| c =
}}
{{eqn | r = x^{1 / n}
| c = {{Defof|Root of Number|$n$th Root}}
}}
{{eqn | ll= \leadsto
| l = \map {f'} x
| r = \frac 1 n x^{\paren {1 / n} - 1}
| c = Power Rule for Derivatives
}}
{{eqn | r = \frac 1 n x^{\paren {1 / n} ... | Let $n \in \N_{>0}$.
Let $f: \R \to \R$ be the [[Definition:Real Function|real function]] defined as $\map f x = \sqrt [n] x$.
Then:
:$\map {f'} x = \dfrac 1 {n \paren {\sqrt [n] x}^{n - 1} }$
everywhere that $\map f x = \sqrt [n] x$ is defined. | {{begin-eqn}}
{{eqn | l = \map f x
| r = \sqrt [n] x
| c =
}}
{{eqn | r = x^{1 / n}
| c = {{Defof|Root of Number|$n$th Root}}
}}
{{eqn | ll= \leadsto
| l = \map {f'} x
| r = \frac 1 n x^{\paren {1 / n} - 1}
| c = [[Power Rule for Derivatives]]
}}
{{eqn | r = \frac 1 n x^{\paren {1 /... | Derivative of Nth Root | https://proofwiki.org/wiki/Derivative_of_Nth_Root | https://proofwiki.org/wiki/Derivative_of_Nth_Root | [
"Derivatives"
] | [
"Definition:Real Function"
] | [
"Power Rule for Derivatives",
"Exponent Combination Laws/Negative Power",
"Category:Derivatives"
] |
proofwiki-9102 | Primitive of Function of Nth Root of a x + b | :$\ds \int \map F {\sqrt [n] {a x + b} } \rd x = \frac n a \int u^{n - 1} \map F u \rd u$
where $u = \sqrt [n] {a x + b}$. | {{begin-eqn}}
{{eqn | l = u
| r = \sqrt [n] {a x + b}
| c =
}}
{{eqn | l = u
| r = \paren {a x + b}^{1/n}
| c =
}}
{{eqn | ll= \leadsto
| l = \frac {\d u} {\d x}
| r = \frac 1 {n \paren {\sqrt [n] {a x + b} }^{n - 1} } \map {\frac \d {\d x} } {a x + b}
| c = Chain Rule for ... | :$\ds \int \map F {\sqrt [n] {a x + b} } \rd x = \frac n a \int u^{n - 1} \map F u \rd u$
where $u = \sqrt [n] {a x + b}$. | {{begin-eqn}}
{{eqn | l = u
| r = \sqrt [n] {a x + b}
| c =
}}
{{eqn | l = u
| r = \paren {a x + b}^{1/n}
| c =
}}
{{eqn | ll= \leadsto
| l = \frac {\d u} {\d x}
| r = \frac 1 {n \paren {\sqrt [n] {a x + b} }^{n - 1} } \map {\frac \d {\d x} } {a x + b}
| c = [[Chain Rule fo... | Primitive of Function of Nth Root of a x + b | https://proofwiki.org/wiki/Primitive_of_Function_of_Nth_Root_of_a_x_+_b | https://proofwiki.org/wiki/Primitive_of_Function_of_Nth_Root_of_a_x_+_b | [
"Integral Substitutions"
] | [] | [
"Derivative of Composite Function",
"Derivative of Nth Root",
"Derivative of Function of Constant Multiple/Corollary",
"Primitive of Composite Function",
"Primitive of Constant Multiple of Function"
] |
proofwiki-9103 | Primitive of Function of Root of a squared minus x squared | :$\ds \int \map F {\sqrt {a^2 - x^2} } \rd x = a \int \map F {a \cos u} \cos u \rd u$
where $x = a \sin u$. | First note that:
{{begin-eqn}}
{{eqn | l = x
| r = a \sin u
| c =
}}
{{eqn | ll= \leadsto
| l = \sqrt {a^2 - x^2}
| r = \sqrt {a^2 - \paren {a \sin u}^2}
| c =
}}
{{eqn | r = a \sqrt {1 - \sin^2 u}
| c = taking $a$ outside the square root
}}
{{eqn | n = 1
| r = a \cos u
... | :$\ds \int \map F {\sqrt {a^2 - x^2} } \rd x = a \int \map F {a \cos u} \cos u \rd u$
where $x = a \sin u$. | First note that:
{{begin-eqn}}
{{eqn | l = x
| r = a \sin u
| c =
}}
{{eqn | ll= \leadsto
| l = \sqrt {a^2 - x^2}
| r = \sqrt {a^2 - \paren {a \sin u}^2}
| c =
}}
{{eqn | r = a \sqrt {1 - \sin^2 u}
| c = taking $a$ outside the square root
}}
{{eqn | n = 1
| r = a \cos u
... | Primitive of Function of Root of a squared minus x squared | https://proofwiki.org/wiki/Primitive_of_Function_of_Root_of_a_squared_minus_x_squared | https://proofwiki.org/wiki/Primitive_of_Function_of_Root_of_a_squared_minus_x_squared | [
"Integral Substitutions"
] | [] | [
"Sum of Squares of Sine and Cosine",
"Derivative of Sine Function",
"Integration by Substitution",
"Primitive of Constant Multiple of Function"
] |
proofwiki-9104 | Primitive of Function of Root of a squared plus x squared | :$\ds \int \map F {\sqrt {a^2 + x^2} } \rd x = a \int \map F {a \sec u} \sec^2 u \rd u$
where $x = a \tan u$. | First note that:
{{begin-eqn}}
{{eqn | l = x
| r = a \tan u
| c =
}}
{{eqn | ll= \leadsto
| l = \sqrt {a^2 + x^2}
| r = \sqrt {a^2 + \paren {a \tan u}^2}
| c =
}}
{{eqn | r = a \sqrt {1 + \tan^2 u}
| c = taking $a$ outside the square root
}}
{{eqn | r = a \sqrt {\sec^2 u}
| c... | :$\ds \int \map F {\sqrt {a^2 + x^2} } \rd x = a \int \map F {a \sec u} \sec^2 u \rd u$
where $x = a \tan u$. | First note that:
{{begin-eqn}}
{{eqn | l = x
| r = a \tan u
| c =
}}
{{eqn | ll= \leadsto
| l = \sqrt {a^2 + x^2}
| r = \sqrt {a^2 + \paren {a \tan u}^2}
| c =
}}
{{eqn | r = a \sqrt {1 + \tan^2 u}
| c = taking $a$ outside the square root
}}
{{eqn | r = a \sqrt {\sec^2 u}
| ... | Primitive of Function of Root of a squared plus x squared | https://proofwiki.org/wiki/Primitive_of_Function_of_Root_of_a_squared_plus_x_squared | https://proofwiki.org/wiki/Primitive_of_Function_of_Root_of_a_squared_plus_x_squared | [
"Integral Substitutions"
] | [] | [
"Sum of Squares of Sine and Cosine/Corollary 1",
"Derivative of Tangent Function",
"Integration by Substitution",
"Primitive of Constant Multiple of Function"
] |
proofwiki-9105 | Primitive of Function of Root of x squared minus a squared | :$\ds \int \map F {\sqrt {x^2 - a^2} } \rd x = a \int \map F {a \tan u} \sec u \tan u \rd u$
where $x = a \sec u$. | First note that:
{{begin-eqn}}
{{eqn | l = x
| r = a \sec u
| c =
}}
{{eqn | ll= \leadsto
| l = \sqrt {x^2 - a^2}
| r = \sqrt {\paren {a \sec u}^2 - a^2}
| c =
}}
{{eqn | r = a \sqrt {\sec^2 u - 1}
| c = taking $a$ outside the square root
}}
{{eqn | r = a \sqrt {\tan^2 u}
| c... | :$\ds \int \map F {\sqrt {x^2 - a^2} } \rd x = a \int \map F {a \tan u} \sec u \tan u \rd u$
where $x = a \sec u$. | First note that:
{{begin-eqn}}
{{eqn | l = x
| r = a \sec u
| c =
}}
{{eqn | ll= \leadsto
| l = \sqrt {x^2 - a^2}
| r = \sqrt {\paren {a \sec u}^2 - a^2}
| c =
}}
{{eqn | r = a \sqrt {\sec^2 u - 1}
| c = taking $a$ outside the square root
}}
{{eqn | r = a \sqrt {\tan^2 u}
| ... | Primitive of Function of Root of x squared minus a squared | https://proofwiki.org/wiki/Primitive_of_Function_of_Root_of_x_squared_minus_a_squared | https://proofwiki.org/wiki/Primitive_of_Function_of_Root_of_x_squared_minus_a_squared | [
"Integral Substitutions"
] | [] | [
"Sum of Squares of Sine and Cosine/Corollary 1",
"Derivative of Secant Function",
"Integration by Substitution",
"Primitive of Constant Multiple of Function"
] |
proofwiki-9106 | Primitive of Function of Exponential Function | :$\ds \int \map F {e^{a x} } \rd x = \frac 1 a \int \frac {\map F u} u \rd u$
where $u = e^{a x}$. | {{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 Exponential of a x
}}
{{eqn | r = a u
| c = Definition of $u$
}}
{{eqn | ll= \leadsto
| l = \int \map F {e^{a x} } \rd x
| r = \int \fra... | :$\ds \int \map F {e^{a x} } \rd x = \frac 1 a \int \frac {\map F u} u \rd u$
where $u = e^{a x}$. | {{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 Exponential of a x]]
}}
{{eqn | r = a u
| c = Definition of $u$
}}
{{eqn | ll= \leadsto
| l = \int \map F {e^{a x} } \rd x
| r = \int ... | Primitive of Function of Exponential Function | https://proofwiki.org/wiki/Primitive_of_Function_of_Exponential_Function | https://proofwiki.org/wiki/Primitive_of_Function_of_Exponential_Function | [
"Integral Substitutions"
] | [] | [
"Derivative of Exponential Function/Corollary 1",
"Primitive of Composite Function",
"Primitive of Constant Multiple of Function"
] |
proofwiki-9107 | Primitive of Function of Natural Logarithm | :$\ds \int \map F {\ln x} \rd x = \int \map F u e^u \rd u$
where $u = \ln x$. | {{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 Function
}}
{{eqn | ll= \leadsto
| l = \int \map F {\ln x} \rd x
| r = \int \map F u x \rd u
| c = Primitive of Composit... | :$\ds \int \map F {\ln x} \rd x = \int \map F u e^u \rd u$
where $u = \ln x$. | {{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 Function]]
}}
{{eqn | ll= \leadsto
| l = \int \map F {\ln x} \rd x
| r = \int \map F u x \rd u
| c = [[Primitive of Co... | Primitive of Function of Natural Logarithm | https://proofwiki.org/wiki/Primitive_of_Function_of_Natural_Logarithm | https://proofwiki.org/wiki/Primitive_of_Function_of_Natural_Logarithm | [
"Integral Substitutions"
] | [] | [
"Derivative of Natural Logarithm Function",
"Primitive of Composite Function"
] |
proofwiki-9108 | Primitive of Function of Arcsine | :$\ds \int \map F {\arcsin \frac x a} \rd x = a \int \map F u \cos u \rd u$
where $u = \arcsin \dfrac x a$. | First note that:
{{begin-eqn}}
{{eqn | l = u
| r = \arcsin \frac x a
| c =
}}
{{eqn | ll= \leadsto
| l = x
| r = a \sin u
| c = {{Defof|Real Arcsine}}
}}
{{end-eqn}}
Then:
{{begin-eqn}}
{{eqn | l = u
| r = \arcsin \frac x a
| c =
}}
{{eqn | ll= \leadsto
| l = \frac {\d... | :$\ds \int \map F {\arcsin \frac x a} \rd x = a \int \map F u \cos u \rd u$
where $u = \arcsin \dfrac x a$. | First note that:
{{begin-eqn}}
{{eqn | l = u
| r = \arcsin \frac x a
| c =
}}
{{eqn | ll= \leadsto
| l = x
| r = a \sin u
| c = {{Defof|Real Arcsine}}
}}
{{end-eqn}}
Then:
{{begin-eqn}}
{{eqn | l = u
| r = \arcsin \frac x a
| c =
}}
{{eqn | ll= \leadsto
| l = \frac ... | Primitive of Function of Arcsine | https://proofwiki.org/wiki/Primitive_of_Function_of_Arcsine | https://proofwiki.org/wiki/Primitive_of_Function_of_Arcsine | [
"Integral Substitutions",
"Arcsine Function",
"Primitives involving Inverse Sine Function"
] | [] | [
"Derivative of Arcsine Function/Corollary",
"Primitive of Composite Function",
"Sum of Squares of Sine and Cosine",
"Primitive of Constant Multiple of Function"
] |
proofwiki-9109 | Primitive of Function of Arccosine | :$\ds \int \map F {\arccos \frac x a} \rd x = -a \int \map F u \sin u \rd u$
where $u = \arccos \dfrac x a$. | First note that:
{{begin-eqn}}
{{eqn | l = u
| r = \arccos \frac x a
| c =
}}
{{eqn | ll= \leadsto
| l = x
| r = a \cos u
| c = {{Defof|Real Arccosine}}
}}
{{end-eqn}}
Then:
{{begin-eqn}}
{{eqn | l = u
| r = \arccos \frac x a
| c =
}}
{{eqn | ll= \leadsto
| l = \frac {... | :$\ds \int \map F {\arccos \frac x a} \rd x = -a \int \map F u \sin u \rd u$
where $u = \arccos \dfrac x a$. | First note that:
{{begin-eqn}}
{{eqn | l = u
| r = \arccos \frac x a
| c =
}}
{{eqn | ll= \leadsto
| l = x
| r = a \cos u
| c = {{Defof|Real Arccosine}}
}}
{{end-eqn}}
Then:
{{begin-eqn}}
{{eqn | l = u
| r = \arccos \frac x a
| c =
}}
{{eqn | ll= \leadsto
| l = \fra... | Primitive of Function of Arccosine | https://proofwiki.org/wiki/Primitive_of_Function_of_Arccosine | https://proofwiki.org/wiki/Primitive_of_Function_of_Arccosine | [
"Integral Substitutions",
"Arccosine Function",
"Primitives involving Inverse Cosine Function"
] | [] | [
"Derivative of Arccosine Function/Corollary",
"Primitive of Composite Function",
"Sum of Squares of Sine and Cosine",
"Primitive of Constant Multiple of Function"
] |
proofwiki-9110 | Primitive of Function of Arctangent | :$\ds \int \map F {\arctan \frac x a} \rd x = a \int \map F u \sec^2 u \rd u$
where $u = \arctan \dfrac x a$. | First note that:
{{begin-eqn}}
{{eqn | l = u
| r = \arctan \frac x a
| c =
}}
{{eqn | ll= \leadsto
| l = x
| r = a \tan u
| c = {{Defof|Real Arctangent}}
}}
{{end-eqn}}
Then:
{{begin-eqn}}
{{eqn | l = u
| r = \arctan \frac x a
| c =
}}
{{eqn | ll= \leadsto
| l = \frac ... | :$\ds \int \map F {\arctan \frac x a} \rd x = a \int \map F u \sec^2 u \rd u$
where $u = \arctan \dfrac x a$. | First note that:
{{begin-eqn}}
{{eqn | l = u
| r = \arctan \frac x a
| c =
}}
{{eqn | ll= \leadsto
| l = x
| r = a \tan u
| c = {{Defof|Real Arctangent}}
}}
{{end-eqn}}
Then:
{{begin-eqn}}
{{eqn | l = u
| r = \arctan \frac x a
| c =
}}
{{eqn | ll= \leadsto
| l = \fr... | Primitive of Function of Arctangent | https://proofwiki.org/wiki/Primitive_of_Function_of_Arctangent | https://proofwiki.org/wiki/Primitive_of_Function_of_Arctangent | [
"Integral Substitutions",
"Arctangent Function",
"Primitives involving Inverse Tangent Function"
] | [] | [
"Derivative of Arctangent Function/Corollary",
"Primitive of Composite Function",
"Sum of Squares of Sine and Cosine/Corollary 1",
"Primitive of Constant Multiple of Function"
] |
proofwiki-9111 | Primitive of Function of Arccotangent | :$\ds \int \map F {\arccot \frac x a} \rd x = -a \int \map F u \csc^2 u \rd u$
where $u = \arccot \dfrac x a$. | First note that:
{{begin-eqn}}
{{eqn | l = u
| r = \arccot \frac x a
| c =
}}
{{eqn | ll= \leadsto
| l = x
| r = a \cot u
| c = {{Defof|Arccotangent}}
}}
{{end-eqn}}
Then:
{{begin-eqn}}
{{eqn | l = u
| r = \arccot \frac x a
| c =
}}
{{eqn | ll= \leadsto
| l = \frac {\d... | :$\ds \int \map F {\arccot \frac x a} \rd x = -a \int \map F u \csc^2 u \rd u$
where $u = \arccot \dfrac x a$. | First note that:
{{begin-eqn}}
{{eqn | l = u
| r = \arccot \frac x a
| c =
}}
{{eqn | ll= \leadsto
| l = x
| r = a \cot u
| c = {{Defof|Arccotangent}}
}}
{{end-eqn}}
Then:
{{begin-eqn}}
{{eqn | l = u
| r = \arccot \frac x a
| c =
}}
{{eqn | ll= \leadsto
| l = \frac ... | Primitive of Function of Arccotangent | https://proofwiki.org/wiki/Primitive_of_Function_of_Arccotangent | https://proofwiki.org/wiki/Primitive_of_Function_of_Arccotangent | [
"Integral Substitutions",
"Arccotangent Function",
"Primitives involving Inverse Cotangent Function"
] | [] | [
"Derivative of Arccotangent Function/Corollary",
"Primitive of Composite Function",
"Sum of Squares of Sine and Cosine/Corollary 2",
"Primitive of Constant Multiple of Function"
] |
proofwiki-9112 | Primitive of Function of Arcsecant | :$\ds \int \map F {\arcsec \frac x a} \rd x = a \int \map F u \sec u \tan u \rd u$
where $u = \arcsec \dfrac x a$. | First note that:
{{begin-eqn}}
{{eqn | l = u
| r = \arcsec \frac x a
| c =
}}
{{eqn | ll= \leadsto
| l = x
| r = a \sec u
| c = {{Defof|Arcsecant}}
}}
{{end-eqn}}
Then:
{{begin-eqn}}
{{eqn | l = u
| r = \arcsec \frac x a
| c =
}}
{{eqn | ll= \leadsto
| l = \frac {\d u}... | :$\ds \int \map F {\arcsec \frac x a} \rd x = a \int \map F u \sec u \tan u \rd u$
where $u = \arcsec \dfrac x a$. | First note that:
{{begin-eqn}}
{{eqn | l = u
| r = \arcsec \frac x a
| c =
}}
{{eqn | ll= \leadsto
| l = x
| r = a \sec u
| c = {{Defof|Arcsecant}}
}}
{{end-eqn}}
Then:
{{begin-eqn}}
{{eqn | l = u
| r = \arcsec \frac x a
| c =
}}
{{eqn | ll= \leadsto
| l = \frac {\d... | Primitive of Function of Arcsecant | https://proofwiki.org/wiki/Primitive_of_Function_of_Arcsecant | https://proofwiki.org/wiki/Primitive_of_Function_of_Arcsecant | [
"Integral Substitutions",
"Arcsecant Function",
"Primitives involving Inverse Secant Function"
] | [] | [
"Derivative of Arcsecant Function/Corollary 1",
"Primitive of Composite Function",
"Sum of Squares of Sine and Cosine/Corollary 1",
"Primitive of Constant Multiple of Function"
] |
proofwiki-9113 | Primitive of Function of Arccosecant | :$\ds \int \map F {\arccsc \frac x a} \rd x = -a \int \map F u \size {\csc u} \cot u \rd u$
where $u = \arccsc \dfrac x a$. | First note that:
{{begin-eqn}}
{{eqn | l = u
| r = \arccsc \frac x a
| c =
}}
{{eqn | ll= \leadsto
| l = x
| r = a \csc u
| c = {{Defof|Arccosecant}}
}}
{{end-eqn}}
Then:
{{begin-eqn}}
{{eqn | l = u
| r = \arccsc \frac x a
| c =
}}
{{eqn | ll= \leadsto
| l = \frac {\d ... | :$\ds \int \map F {\arccsc \frac x a} \rd x = -a \int \map F u \size {\csc u} \cot u \rd u$
where $u = \arccsc \dfrac x a$. | First note that:
{{begin-eqn}}
{{eqn | l = u
| r = \arccsc \frac x a
| c =
}}
{{eqn | ll= \leadsto
| l = x
| r = a \csc u
| c = {{Defof|Arccosecant}}
}}
{{end-eqn}}
Then:
{{begin-eqn}}
{{eqn | l = u
| r = \arccsc \frac x a
| c =
}}
{{eqn | ll= \leadsto
| l = \frac {... | Primitive of Function of Arccosecant | https://proofwiki.org/wiki/Primitive_of_Function_of_Arccosecant | https://proofwiki.org/wiki/Primitive_of_Function_of_Arccosecant | [
"Integral Substitutions",
"Arccosecant Function",
"Primitives involving Inverse Cosecant Function"
] | [] | [
"Derivative of Arccosecant Function/Corollary",
"Primitive of Composite Function",
"Sum of Squares of Sine and Cosine/Corollary 2",
"Primitive of Constant Multiple of Function"
] |
proofwiki-9114 | Primitive of Reciprocal of a x + b | :$\ds \int \frac {\d x} {a x + b} = \frac 1 a \ln \size {a x + b} + C$
where $a \ne 0$ and $x \ne - \dfrac b a$. | {{begin-eqn}}
{{eqn | l = \int \frac {\d x} {a x + b}
| r = \frac 1 a \int \frac {\map \d {a x + b} } {a x + b}
| c = Primitive of Function of $a x + b$
}}
{{eqn | r = \frac 1 a \ln \size {a x + b} + C
| c = Primitive of Reciprocal
}}
{{end-eqn}}
{{qed}} | :$\ds \int \frac {\d x} {a x + b} = \frac 1 a \ln \size {a x + b} + C$
where $a \ne 0$ and $x \ne - \dfrac b a$. | {{begin-eqn}}
{{eqn | l = \int \frac {\d x} {a x + b}
| r = \frac 1 a \int \frac {\map \d {a x + b} } {a x + b}
| c = [[Primitive of Function of a x + b|Primitive of Function of $a x + b$]]
}}
{{eqn | r = \frac 1 a \ln \size {a x + b} + C
| c = [[Primitive of Reciprocal]]
}}
{{end-eqn}}
{{qed}} | Primitive of Reciprocal of a x + b | https://proofwiki.org/wiki/Primitive_of_Reciprocal_of_a_x_+_b | https://proofwiki.org/wiki/Primitive_of_Reciprocal_of_a_x_+_b | [
"Primitive of Reciprocal of a x + b",
"Primitives involving a x + b",
"Primitives involving Reciprocals"
] | [] | [
"Primitive of Function of a x + b",
"Primitive of Reciprocal"
] |
proofwiki-9115 | Primitive of Reciprocal of a x + b | :$\ds \int \frac {\d x} {a x + b} = \frac 1 a \ln \size {a x + b} + C$
where $a \ne 0$ and $x \ne - \dfrac b a$. | Let $u = a x + b$.
Then:
{{begin-eqn}}
{{eqn | l = \int \frac {\rd x} {\paren {a x + b}^3}
| r = \frac 1 a \int \frac {\rd u} {u^3}
| c = Primitive of Function of $a x + b$
}}
{{eqn | r = \frac 1 a \frac {-1} {2 u^2} + C
| c = Primitive of Power
}}
{{eqn | r = \frac {-1} {2 a \paren {a x + b}^2} + C
... | :$\ds \int \frac {\d x} {a x + b} = \frac 1 a \ln \size {a x + b} + C$
where $a \ne 0$ and $x \ne - \dfrac b a$. | Let $u = a x + b$.
Then:
{{begin-eqn}}
{{eqn | l = \int \frac {\rd x} {\paren {a x + b}^3}
| r = \frac 1 a \int \frac {\rd u} {u^3}
| c = [[Primitive of Function of a x + b|Primitive of Function of $a x + b$]]
}}
{{eqn | r = \frac 1 a \frac {-1} {2 u^2} + C
| c = [[Primitive of Power]]
}}
{{eqn | r ... | Primitive of Reciprocal of a x + b cubed/Proof 1 | https://proofwiki.org/wiki/Primitive_of_Reciprocal_of_a_x_+_b | https://proofwiki.org/wiki/Primitive_of_Reciprocal_of_a_x_+_b_cubed/Proof_1 | [
"Primitive of Reciprocal of a x + b",
"Primitives involving a x + b",
"Primitives involving Reciprocals"
] | [] | [
"Primitive of Function of a x + b",
"Primitive of Power"
] |
proofwiki-9116 | Primitive of Reciprocal of a x + b | :$\ds \int \frac {\d x} {a x + b} = \frac 1 a \ln \size {a x + b} + C$
where $a \ne 0$ and $x \ne - \dfrac b a$. | From Primitive of Power of $a x + b$:
:$\ds \int \paren {a x + b}^n \rd x = \frac {\paren {a x + b}^{n + 1} } {\paren {n + 1} a} + C$
where $n \ne 1$.
The result follows by setting $n = -3$.
{{qed}} | :$\ds \int \frac {\d x} {a x + b} = \frac 1 a \ln \size {a x + b} + C$
where $a \ne 0$ and $x \ne - \dfrac b a$. | From [[Primitive of Power of a x + b|Primitive of Power of $a x + b$]]:
:$\ds \int \paren {a x + b}^n \rd x = \frac {\paren {a x + b}^{n + 1} } {\paren {n + 1} a} + C$
where $n \ne 1$.
The result follows by setting $n = -3$.
{{qed}} | Primitive of Reciprocal of a x + b cubed/Proof 2 | https://proofwiki.org/wiki/Primitive_of_Reciprocal_of_a_x_+_b | https://proofwiki.org/wiki/Primitive_of_Reciprocal_of_a_x_+_b_cubed/Proof_2 | [
"Primitive of Reciprocal of a x + b",
"Primitives involving a x + b",
"Primitives involving Reciprocals"
] | [] | [
"Primitive of Power of a x + b"
] |
proofwiki-9117 | Primitive of Reciprocal of a x + b | :$\ds \int \frac {\d x} {a x + b} = \frac 1 a \ln \size {a x + b} + C$
where $a \ne 0$ and $x \ne - \dfrac b a$. | Let $u = a x + b$.
Then:
{{begin-eqn}}
{{eqn | l = \int \frac {\d x} {\paren {a x + b}^2}
| r = \frac 1 a \int \frac {\d u} {u^2}
| c = Primitive of Function of $a x + b$
}}
{{eqn | r = \frac 1 a \frac {-1} u + C
| c = Primitive of Power
}}
{{eqn | r = -\frac 1 {a \paren {a x + b} } + C
| c = su... | :$\ds \int \frac {\d x} {a x + b} = \frac 1 a \ln \size {a x + b} + C$
where $a \ne 0$ and $x \ne - \dfrac b a$. | Let $u = a x + b$.
Then:
{{begin-eqn}}
{{eqn | l = \int \frac {\d x} {\paren {a x + b}^2}
| r = \frac 1 a \int \frac {\d u} {u^2}
| c = [[Primitive of Function of a x + b|Primitive of Function of $a x + b$]]
}}
{{eqn | r = \frac 1 a \frac {-1} u + C
| c = [[Primitive of Power]]
}}
{{eqn | r = -\frac... | Primitive of Reciprocal of a x + b squared/Proof 1 | https://proofwiki.org/wiki/Primitive_of_Reciprocal_of_a_x_+_b | https://proofwiki.org/wiki/Primitive_of_Reciprocal_of_a_x_+_b_squared/Proof_1 | [
"Primitive of Reciprocal of a x + b",
"Primitives involving a x + b",
"Primitives involving Reciprocals"
] | [] | [
"Primitive of Function of a x + b",
"Primitive of Power"
] |
proofwiki-9118 | Primitive of Reciprocal of a x + b | :$\ds \int \frac {\d x} {a x + b} = \frac 1 a \ln \size {a x + b} + C$
where $a \ne 0$ and $x \ne - \dfrac b a$. | From Primitive of Power of $a x + b$:
:$\ds \int \paren {a x + b}^n \rd x = \frac {\paren {a x + b}^{n + 1} } {\paren {n + 1} a} + C$
where $n \ne 1$.
The result follows by setting $n = -2$.
{{qed}} | :$\ds \int \frac {\d x} {a x + b} = \frac 1 a \ln \size {a x + b} + C$
where $a \ne 0$ and $x \ne - \dfrac b a$. | From [[Primitive of Power of a x + b|Primitive of Power of $a x + b$]]:
:$\ds \int \paren {a x + b}^n \rd x = \frac {\paren {a x + b}^{n + 1} } {\paren {n + 1} a} + C$
where $n \ne 1$.
The result follows by setting $n = -2$.
{{qed}} | Primitive of Reciprocal of a x + b squared/Proof 2 | https://proofwiki.org/wiki/Primitive_of_Reciprocal_of_a_x_+_b | https://proofwiki.org/wiki/Primitive_of_Reciprocal_of_a_x_+_b_squared/Proof_2 | [
"Primitive of Reciprocal of a x + b",
"Primitives involving a x + b",
"Primitives involving Reciprocals"
] | [] | [
"Primitive of Power of a x + b"
] |
proofwiki-9119 | Primitive of x over a x + b | :$\ds \int \frac {x \rd x} {a x + b} = \frac x a - \frac b {a^2} \ln \size {a x + b} + C$ | Put $u = a x + b$.
Then:
{{begin-eqn}}
{{eqn | l = x
| r = \frac {u - b} a
| c =
}}
{{eqn | l = \frac {\d u} {\d x}
| r = \frac 1 a
| c =
}}
{{end-eqn}}
Then:
{{begin-eqn}}
{{eqn | l = \int \frac {x \rd x} {\paren {a x + b}^3}
| r = \int \frac 1 a \frac {u - b} {a u^3} \rd u
| c = ... | :$\ds \int \frac {x \rd x} {a x + b} = \frac x a - \frac b {a^2} \ln \size {a x + b} + C$ | Put $u = a x + b$.
Then:
{{begin-eqn}}
{{eqn | l = x
| r = \frac {u - b} a
| c =
}}
{{eqn | l = \frac {\d u} {\d x}
| r = \frac 1 a
| c =
}}
{{end-eqn}}
Then:
{{begin-eqn}}
{{eqn | l = \int \frac {x \rd x} {\paren {a x + b}^3}
| r = \int \frac 1 a \frac {u - b} {a u^3} \rd u
| ... | Primitive of x over a x + b cubed/Proof 1 | https://proofwiki.org/wiki/Primitive_of_x_over_a_x_+_b | https://proofwiki.org/wiki/Primitive_of_x_over_a_x_+_b_cubed/Proof_1 | [
"Primitive of x over a x + b",
"Primitives involving a x + b"
] | [] | [
"Integration by Substitution",
"Linear Combination of Integrals/Indefinite",
"Primitive of Power"
] |
proofwiki-9120 | Primitive of x over a x + b | :$\ds \int \frac {x \rd x} {a x + b} = \frac x a - \frac b {a^2} \ln \size {a x + b} + C$ | From Primitive of $x$ by Power of $a x + b$:
:$\ds \int x \paren {a x + b}^n \rd x = \frac {\paren {a x + b}^{n + 2} } {\paren {n + 2} a^2} - \frac {b \paren {a x + b}^{n + 1} } {\paren {n + 1} a^2} + C$
where $n \ne - 1$ and $n \ne - 2$.
The result follows by setting $n = -3$.
{{qed}} | :$\ds \int \frac {x \rd x} {a x + b} = \frac x a - \frac b {a^2} \ln \size {a x + b} + C$ | From [[Primitive of x by Power of a x + b|Primitive of $x$ by Power of $a x + b$]]:
:$\ds \int x \paren {a x + b}^n \rd x = \frac {\paren {a x + b}^{n + 2} } {\paren {n + 2} a^2} - \frac {b \paren {a x + b}^{n + 1} } {\paren {n + 1} a^2} + C$
where $n \ne - 1$ and $n \ne - 2$.
The result follows by setting $n = -3$.
... | Primitive of x over a x + b cubed/Proof 2 | https://proofwiki.org/wiki/Primitive_of_x_over_a_x_+_b | https://proofwiki.org/wiki/Primitive_of_x_over_a_x_+_b_cubed/Proof_2 | [
"Primitive of x over a x + b",
"Primitives involving a x + b"
] | [] | [
"Primitive of x by Power of a x + b"
] |
proofwiki-9121 | Primitive of x over a x + b | :$\ds \int \frac {x \rd x} {a x + b} = \frac x a - \frac b {a^2} \ln \size {a x + b} + C$ | Put $u = a x + b$.
Then:
{{begin-eqn}}
{{eqn | l = x
| r = \frac {u - b} a
| c =
}}
{{eqn | l = \frac {\d u} {\d x}
| r = \frac 1 a
| c =
}}
{{end-eqn}}
Then:
{{begin-eqn}}
{{eqn | l = \int \frac {x \rd x} {\paren {a x + b}^2}
| r = \int \frac 1 a \frac {u - b} {a u^2} \rd u
| c = ... | :$\ds \int \frac {x \rd x} {a x + b} = \frac x a - \frac b {a^2} \ln \size {a x + b} + C$ | Put $u = a x + b$.
Then:
{{begin-eqn}}
{{eqn | l = x
| r = \frac {u - b} a
| c =
}}
{{eqn | l = \frac {\d u} {\d x}
| r = \frac 1 a
| c =
}}
{{end-eqn}}
Then:
{{begin-eqn}}
{{eqn | l = \int \frac {x \rd x} {\paren {a x + b}^2}
| r = \int \frac 1 a \frac {u - b} {a u^2} \rd u
| ... | Primitive of x over a x + b squared/Proof 1 | https://proofwiki.org/wiki/Primitive_of_x_over_a_x_+_b | https://proofwiki.org/wiki/Primitive_of_x_over_a_x_+_b_squared/Proof_1 | [
"Primitive of x over a x + b",
"Primitives involving a x + b"
] | [] | [
"Integration by Substitution",
"Linear Combination of Integrals/Indefinite",
"Primitive of Reciprocal",
"Primitive of Power"
] |
proofwiki-9122 | Primitive of x over a x + b | :$\ds \int \frac {x \rd x} {a x + b} = \frac x a - \frac b {a^2} \ln \size {a x + b} + C$ | {{begin-eqn}}
{{eqn | l = \int \frac {x \rd x} {\paren {a x + b}^2}
| r = \int \frac {a x \rd x} {a \paren {a x + b}^2}
| c = multiplying top and bottom by $a$
}}
{{eqn | r = \int \frac {\paren {a x + b - b} \rd x} {a \paren {a x + b}^2}
| c = adding and subtracting $b$
}}
{{eqn | r = \frac 1 a \int \... | :$\ds \int \frac {x \rd x} {a x + b} = \frac x a - \frac b {a^2} \ln \size {a x + b} + C$ | {{begin-eqn}}
{{eqn | l = \int \frac {x \rd x} {\paren {a x + b}^2}
| r = \int \frac {a x \rd x} {a \paren {a x + b}^2}
| c = multiplying [[Definition:Numerator|top]] and [[Definition:Denominator|bottom]] by $a$
}}
{{eqn | r = \int \frac {\paren {a x + b - b} \rd x} {a \paren {a x + b}^2}
| c = adding... | Primitive of x over a x + b squared/Proof 2 | https://proofwiki.org/wiki/Primitive_of_x_over_a_x_+_b | https://proofwiki.org/wiki/Primitive_of_x_over_a_x_+_b_squared/Proof_2 | [
"Primitive of x over a x + b",
"Primitives involving a x + b"
] | [] | [
"Definition:Fraction/Numerator",
"Definition:Fraction/Denominator",
"Linear Combination of Integrals/Indefinite",
"Primitive of Reciprocal of a x + b",
"Primitive of Reciprocal of a x + b squared"
] |
proofwiki-9123 | Primitive of x over a x + b | :$\ds \int \frac {x \rd x} {a x + b} = \frac x a - \frac b {a^2} \ln \size {a x + b} + C$ | Put $u = a x + b$
Then:
{{begin-eqn}}
{{eqn | l = x
| r = \frac {u - b} a
| c =
}}
{{eqn | l = \frac {\d x} {\d u}
| r = \frac 1 a
| c =
}}
{{end-eqn}}
Then:
{{begin-eqn}}
{{eqn | l = \int \frac {x \rd x} {a x + b}
| r = \int \frac 1 a \frac {u - b} {a u} \rd u
| c = Integration by... | :$\ds \int \frac {x \rd x} {a x + b} = \frac x a - \frac b {a^2} \ln \size {a x + b} + C$ | Put $u = a x + b$
Then:
{{begin-eqn}}
{{eqn | l = x
| r = \frac {u - b} a
| c =
}}
{{eqn | l = \frac {\d x} {\d u}
| r = \frac 1 a
| c =
}}
{{end-eqn}}
Then:
{{begin-eqn}}
{{eqn | l = \int \frac {x \rd x} {a x + b}
| r = \int \frac 1 a \frac {u - b} {a u} \rd u
| c = [[Integrat... | Primitive of x over a x + b/Proof 1 | https://proofwiki.org/wiki/Primitive_of_x_over_a_x_+_b | https://proofwiki.org/wiki/Primitive_of_x_over_a_x_+_b/Proof_1 | [
"Primitive of x over a x + b",
"Primitives involving a x + b"
] | [] | [
"Integration by Substitution",
"Linear Combination of Integrals/Indefinite",
"Primitive of Constant",
"Primitive of Reciprocal",
"Definition:Primitive (Calculus)/Constant of Integration"
] |
proofwiki-9124 | Primitive of x over a x + b | :$\ds \int \frac {x \rd x} {a x + b} = \frac x a - \frac b {a^2} \ln \size {a x + b} + C$ | From Primitive of Power of $x$ by Power of $a x + b$: Decrement of Power of $x$:
:$\ds \int x^m \paren {a x + b}^n \rd x = \frac {x^m \paren {a x + b}^{n + 1} } {\paren {m + n + 1} a} - \frac {m b} {\paren {m + n + 1} a} \int x^{m - 1} \paren {a x + b}^n \rd x$
Let $m = 1$ and $n = -1$.
Then:
{{begin-eqn}}
{{eqn | l = ... | :$\ds \int \frac {x \rd x} {a x + b} = \frac x a - \frac b {a^2} \ln \size {a x + b} + C$ | From [[Primitive of Power of x by Power of a x + b/Decrement of Power of x|Primitive of Power of $x$ by Power of $a x + b$: Decrement of Power of $x$]]:
:$\ds \int x^m \paren {a x + b}^n \rd x = \frac {x^m \paren {a x + b}^{n + 1} } {\paren {m + n + 1} a} - \frac {m b} {\paren {m + n + 1} a} \int x^{m - 1} \paren {a x ... | Primitive of x over a x + b/Proof 2 | https://proofwiki.org/wiki/Primitive_of_x_over_a_x_+_b | https://proofwiki.org/wiki/Primitive_of_x_over_a_x_+_b/Proof_2 | [
"Primitive of x over a x + b",
"Primitives involving a x + b"
] | [] | [
"Primitive of Power of x by Power of a x + b/Decrement of Power of x",
"Primitive of Reciprocal of a x + b"
] |
proofwiki-9125 | Primitive of x over a x + b | :$\ds \int \frac {x \rd x} {a x + b} = \frac x a - \frac b {a^2} \ln \size {a x + b} + C$ | {{begin-eqn}}
{{eqn | l = \int \frac {x \rd x} {a x + b}
| r = \int \frac 1 a \frac {a x \rd x} {a x + b}
| c =
}}
{{eqn | r = \int \frac 1 a \frac {\paren {a x + b - b} \rd x} {a x + b}
| c =
}}
{{eqn | r = \frac 1 a \int \frac {\paren {a x + b} \rd x} {a x + b} - \frac b a \int \frac {\d x} {a x +... | :$\ds \int \frac {x \rd x} {a x + b} = \frac x a - \frac b {a^2} \ln \size {a x + b} + C$ | {{begin-eqn}}
{{eqn | l = \int \frac {x \rd x} {a x + b}
| r = \int \frac 1 a \frac {a x \rd x} {a x + b}
| c =
}}
{{eqn | r = \int \frac 1 a \frac {\paren {a x + b - b} \rd x} {a x + b}
| c =
}}
{{eqn | r = \frac 1 a \int \frac {\paren {a x + b} \rd x} {a x + b} - \frac b a \int \frac {\d x} {a x +... | Primitive of x over a x + b/Proof 3 | https://proofwiki.org/wiki/Primitive_of_x_over_a_x_+_b | https://proofwiki.org/wiki/Primitive_of_x_over_a_x_+_b/Proof_3 | [
"Primitive of x over a x + b",
"Primitives involving a x + b"
] | [] | [
"Linear Combination of Integrals/Indefinite",
"Primitive of Constant",
"Primitive of Reciprocal of a x + b"
] |
proofwiki-9126 | First Translation Property of Laplace Transforms | :$\laptrans {e^{a t} \map f t} = \map F {s - a}$
everywhere that $\laptrans f$ exists, for $\map \Re s > a$ | {{begin-eqn}}
{{eqn | l = \laptrans {e^{a t} \map f t}
| r = \int_0^{\to +\infty} e^{-s t} e^{a t} \map f t \rd t
| c = {{Defof|Laplace Transform}}
}}
{{eqn | r = \int_0^{\to +\infty} e^{-s t + a t} \map f t \rd t
| c = Exponent Combination Laws
}}
{{eqn | r = \int_0^{\to +\infty} e^{-\paren {s - a} t... | :$\laptrans {e^{a t} \map f t} = \map F {s - a}$
everywhere that $\laptrans f$ exists, for $\map \Re s > a$ | {{begin-eqn}}
{{eqn | l = \laptrans {e^{a t} \map f t}
| r = \int_0^{\to +\infty} e^{-s t} e^{a t} \map f t \rd t
| c = {{Defof|Laplace Transform}}
}}
{{eqn | r = \int_0^{\to +\infty} e^{-s t + a t} \map f t \rd t
| c = [[Exponent Combination Laws]]
}}
{{eqn | r = \int_0^{\to +\infty} e^{-\paren {s - ... | First Translation Property of Laplace Transforms | https://proofwiki.org/wiki/First_Translation_Property_of_Laplace_Transforms | https://proofwiki.org/wiki/First_Translation_Property_of_Laplace_Transforms | [
"First Translation Property of Laplace Transforms",
"Properties of Laplace Transforms",
"Laplace Transforms",
"Exponential Function"
] | [] | [
"Exponent Combination Laws"
] |
proofwiki-9127 | First Translation Property of Laplace Transforms | :$\laptrans {e^{a t} \map f t} = \map F {s - a}$
everywhere that $\laptrans f$ exists, for $\map \Re s > a$ | {{begin-eqn}}
{{eqn | l = \laptrans {\cosh 5 t}
| r = \dfrac 5 {s^2 - 5^2}
| c = Laplace Transform of Hyperbolic Cosine
}}
{{eqn | ll= \leadsto
| l = \laptrans {e^{4 t} \cosh 5 t}
| r = \dfrac {s - 4} {\paren {s - 4}^2 - 25}
| c = First Translation Property of Laplace Transforms
}}
{{eqn |... | :$\laptrans {e^{a t} \map f t} = \map F {s - a}$
everywhere that $\laptrans f$ exists, for $\map \Re s > a$ | {{begin-eqn}}
{{eqn | l = \laptrans {\cosh 5 t}
| r = \dfrac 5 {s^2 - 5^2}
| c = [[Laplace Transform of Hyperbolic Cosine]]
}}
{{eqn | ll= \leadsto
| l = \laptrans {e^{4 t} \cosh 5 t}
| r = \dfrac {s - 4} {\paren {s - 4}^2 - 25}
| c = [[First Translation Property of Laplace Transforms]]
}}... | First Translation Property of Laplace Transforms/Examples/Example 4/Proof 1 | https://proofwiki.org/wiki/First_Translation_Property_of_Laplace_Transforms | https://proofwiki.org/wiki/First_Translation_Property_of_Laplace_Transforms/Examples/Example_4/Proof_1 | [
"First Translation Property of Laplace Transforms",
"Properties of Laplace Transforms",
"Laplace Transforms",
"Exponential Function"
] | [] | [
"Laplace Transform of Hyperbolic Cosine",
"First Translation Property of Laplace Transforms"
] |
proofwiki-9128 | First Translation Property of Laplace Transforms | :$\laptrans {e^{a t} \map f t} = \map F {s - a}$
everywhere that $\laptrans f$ exists, for $\map \Re s > a$ | {{begin-eqn}}
{{eqn | l = \laptrans {e^{4 t} \cosh 5 t}
| r = \laptrans {e^{4 t} \paren {\dfrac {e^{5 t} + e^{-5 t} } 2} }
| c = {{Defof|Hyperbolic Cosine}}
}}
{{eqn | r = \dfrac 1 2 \laptrans {e^{9 t} + e^{-t} }
| c =
}}
{{eqn | r = \dfrac 1 2 \paren {\dfrac 1 {s - 9} + \dfrac 1 {s + 1} }
| c ... | :$\laptrans {e^{a t} \map f t} = \map F {s - a}$
everywhere that $\laptrans f$ exists, for $\map \Re s > a$ | {{begin-eqn}}
{{eqn | l = \laptrans {e^{4 t} \cosh 5 t}
| r = \laptrans {e^{4 t} \paren {\dfrac {e^{5 t} + e^{-5 t} } 2} }
| c = {{Defof|Hyperbolic Cosine}}
}}
{{eqn | r = \dfrac 1 2 \laptrans {e^{9 t} + e^{-t} }
| c =
}}
{{eqn | r = \dfrac 1 2 \paren {\dfrac 1 {s - 9} + \dfrac 1 {s + 1} }
| c ... | First Translation Property of Laplace Transforms/Examples/Example 4/Proof 2 | https://proofwiki.org/wiki/First_Translation_Property_of_Laplace_Transforms | https://proofwiki.org/wiki/First_Translation_Property_of_Laplace_Transforms/Examples/Example_4/Proof_2 | [
"First Translation Property of Laplace Transforms",
"Properties of Laplace Transforms",
"Laplace Transforms",
"Exponential Function"
] | [] | [
"Laplace Transform of Exponential"
] |
proofwiki-9129 | Laplace Transform of Derivative | Let $f'$ be piecewise continuous with one-sided limits on said intervals.
Then $\laptrans f$ exists for $\map \Re s > a$, and:
:$\laptrans {\map {f'} t} = s \laptrans {\map f t} - \map f 0$ | {{begin-eqn}}
{{eqn | l = \laptrans {\map {f'} t}
| r = \int_0^{\mathop \to +\infty} e^{-s t} \map {f'} t \rd t
| c = {{Defof|Laplace Transform}}
}}
{{eqn | r = \lim_{A \mathop \to +\infty} \int_0^A e^{-s t} \map {f'} t \rd t
| c = {{Defof|Improper Integral on Closed Interval Unbounded Above}}
}}
{{en... | Let $f'$ be [[Definition:Piecewise Continuous Function with One-Sided Limits|piecewise continuous with one-sided limits]] on said [[Definition:Real Interval|intervals]].
Then $\laptrans f$ exists for $\map \Re s > a$, and:
:$\laptrans {\map {f'} t} = s \laptrans {\map f t} - \map f 0$ | {{begin-eqn}}
{{eqn | l = \laptrans {\map {f'} t}
| r = \int_0^{\mathop \to +\infty} e^{-s t} \map {f'} t \rd t
| c = {{Defof|Laplace Transform}}
}}
{{eqn | r = \lim_{A \mathop \to +\infty} \int_0^A e^{-s t} \map {f'} t \rd t
| c = {{Defof|Improper Integral on Closed Interval Unbounded Above}}
}}
{{en... | Laplace Transform of Derivative | https://proofwiki.org/wiki/Laplace_Transform_of_Derivative | https://proofwiki.org/wiki/Laplace_Transform_of_Derivative | [
"Laplace Transforms of Derivatives",
"Laplace Transforms",
"Derivatives"
] | [
"Definition:Piecewise Continuous Function/One-Sided Limits",
"Definition:Real Interval"
] | [
"Definition:Piecewise Continuous Function/One-Sided Limits",
"Piecewise Continuous Function with One-Sided Limits is Darboux Integrable",
"Integration by Parts",
"Definition:Exponential Order",
"Complex Modulus of Product of Complex Numbers",
"Exponential Tends to Zero and Infinity",
"Exponent Combinati... |
proofwiki-9130 | Primitive of x squared over a x + b | :$\ds \int \frac {x^2 \rd x} {a x + b} = \frac {\paren {a x + b}^2} {2 a^3} - \frac {2 b \paren {a x + b} } {a^3} + \frac {b^2} {a^3} \ln \size {a x + b} + C$ | Put $u = a x + b$.
Then:
{{begin-eqn}}
{{eqn | l = x
| r = \frac {u - b} a
| c =
}}
{{eqn | l = \frac {\d u} {\d x}
| r = \frac 1 a
| c =
}}
{{end-eqn}}
Then:
{{begin-eqn}}
{{eqn | l = \int \frac {x^2 \rd x} {\paren {a x + b}^2}
| r = \int \frac 1 a \paren {\frac {u - b} a}^2 \frac 1 {u^... | :$\ds \int \frac {x^2 \rd x} {a x + b} = \frac {\paren {a x + b}^2} {2 a^3} - \frac {2 b \paren {a x + b} } {a^3} + \frac {b^2} {a^3} \ln \size {a x + b} + C$ | Put $u = a x + b$.
Then:
{{begin-eqn}}
{{eqn | l = x
| r = \frac {u - b} a
| c =
}}
{{eqn | l = \frac {\d u} {\d x}
| r = \frac 1 a
| c =
}}
{{end-eqn}}
Then:
{{begin-eqn}}
{{eqn | l = \int \frac {x^2 \rd x} {\paren {a x + b}^2}
| r = \int \frac 1 a \paren {\frac {u - b} a}^2 \frac 1... | Primitive of x squared over a x + b squared/Proof 1 | https://proofwiki.org/wiki/Primitive_of_x_squared_over_a_x_+_b | https://proofwiki.org/wiki/Primitive_of_x_squared_over_a_x_+_b_squared/Proof_1 | [
"Primitive of x squared over a x + b",
"Primitives involving a x + b"
] | [] | [
"Integration by Substitution",
"Square of Difference",
"Linear Combination of Integrals/Indefinite",
"Primitive of Constant",
"Primitive of Reciprocal",
"Primitive of Power"
] |
proofwiki-9131 | Primitive of x squared over a x + b | :$\ds \int \frac {x^2 \rd x} {a x + b} = \frac {\paren {a x + b}^2} {2 a^3} - \frac {2 b \paren {a x + b} } {a^3} + \frac {b^2} {a^3} \ln \size {a x + b} + C$ | {{begin-eqn}}
{{eqn | l = \int \frac {x^2 \rd x} {\paren {a x + b}^2}
| r = \int \frac {a x^2 \rd x} {a \paren {a x + b}^2}
| c = multiplying top and bottom by $a$
}}
{{eqn | r = \int \frac {x \paren {a x + b - b} \rd x} {a \paren {a x + b}^2}
| c = adding and subtracting $b x$
}}
{{eqn | r = \frac 1 ... | :$\ds \int \frac {x^2 \rd x} {a x + b} = \frac {\paren {a x + b}^2} {2 a^3} - \frac {2 b \paren {a x + b} } {a^3} + \frac {b^2} {a^3} \ln \size {a x + b} + C$ | {{begin-eqn}}
{{eqn | l = \int \frac {x^2 \rd x} {\paren {a x + b}^2}
| r = \int \frac {a x^2 \rd x} {a \paren {a x + b}^2}
| c = multiplying [[Definition:Numerator|top]] and [[Definition:Denominator|bottom]] by $a$
}}
{{eqn | r = \int \frac {x \paren {a x + b - b} \rd x} {a \paren {a x + b}^2}
| c = ... | Primitive of x squared over a x + b squared/Proof 2 | https://proofwiki.org/wiki/Primitive_of_x_squared_over_a_x_+_b | https://proofwiki.org/wiki/Primitive_of_x_squared_over_a_x_+_b_squared/Proof_2 | [
"Primitive of x squared over a x + b",
"Primitives involving a x + b"
] | [] | [
"Definition:Fraction/Numerator",
"Definition:Fraction/Denominator",
"Linear Combination of Integrals/Indefinite",
"Primitive of x over a x + b",
"Primitive of x over a x + b squared"
] |
proofwiki-9132 | Primitive of x squared over a x + b | :$\ds \int \frac {x^2 \rd x} {a x + b} = \frac {\paren {a x + b}^2} {2 a^3} - \frac {2 b \paren {a x + b} } {a^3} + \frac {b^2} {a^3} \ln \size {a x + b} + C$ | Put $u = a x + b$.
Then:
{{begin-eqn}}
{{eqn | l = x
| r = \frac {u - b} a
| c =
}}
{{eqn | l = \frac {\d u} {\d x}
| r = \frac 1 a
| c =
}}
{{end-eqn}}
Then:
{{begin-eqn}}
{{eqn | l = \int \frac {x^2 \rd x} {a x + b}
| r = \int \frac 1 a \paren {\frac {u - b} a}^2 \frac {\d u} u
|... | :$\ds \int \frac {x^2 \rd x} {a x + b} = \frac {\paren {a x + b}^2} {2 a^3} - \frac {2 b \paren {a x + b} } {a^3} + \frac {b^2} {a^3} \ln \size {a x + b} + C$ | Put $u = a x + b$.
Then:
{{begin-eqn}}
{{eqn | l = x
| r = \frac {u - b} a
| c =
}}
{{eqn | l = \frac {\d u} {\d x}
| r = \frac 1 a
| c =
}}
{{end-eqn}}
Then:
{{begin-eqn}}
{{eqn | l = \int \frac {x^2 \rd x} {a x + b}
| r = \int \frac 1 a \paren {\frac {u - b} a}^2 \frac {\d u} u
... | Primitive of x squared over a x + b/Proof 1 | https://proofwiki.org/wiki/Primitive_of_x_squared_over_a_x_+_b | https://proofwiki.org/wiki/Primitive_of_x_squared_over_a_x_+_b/Proof_1 | [
"Primitive of x squared over a x + b",
"Primitives involving a x + b"
] | [] | [
"Integration by Substitution",
"Primitive of Constant Multiple of Function",
"Linear Combination of Integrals/Indefinite",
"Primitive of Power",
"Primitive of Constant",
"Primitive of Constant Multiple of Function",
"Primitive of Reciprocal",
"Definition:Primitive (Calculus)/Constant of Integration"
] |
proofwiki-9133 | Primitive of x squared over a x + b | :$\ds \int \frac {x^2 \rd x} {a x + b} = \frac {\paren {a x + b}^2} {2 a^3} - \frac {2 b \paren {a x + b} } {a^3} + \frac {b^2} {a^3} \ln \size {a x + b} + C$ | From Primitive of $x^m \paren {a x + b}^n$: Decrement of Power of $x$:
:$\ds \int x^m \paren {a x + b}^n \rd x = \frac {x^m \paren {a x + b}^{n + 1} } {\paren {m + n + 1} a} - \frac {m b} {\paren {m + n + 1} a} \int x^{m - 1} \paren {a x + b}^n \rd x$
Let $m = 2$ and $n = -1$.
Then:
{{begin-eqn}}
{{eqn | l = \int \frac... | :$\ds \int \frac {x^2 \rd x} {a x + b} = \frac {\paren {a x + b}^2} {2 a^3} - \frac {2 b \paren {a x + b} } {a^3} + \frac {b^2} {a^3} \ln \size {a x + b} + C$ | From [[Primitive of Power of x by Power of a x + b/Decrement of Power of x|Primitive of $x^m \paren {a x + b}^n$: Decrement of Power of $x$]]:
:$\ds \int x^m \paren {a x + b}^n \rd x = \frac {x^m \paren {a x + b}^{n + 1} } {\paren {m + n + 1} a} - \frac {m b} {\paren {m + n + 1} a} \int x^{m - 1} \paren {a x + b}^n \rd... | Primitive of x squared over a x + b/Proof 2 | https://proofwiki.org/wiki/Primitive_of_x_squared_over_a_x_+_b | https://proofwiki.org/wiki/Primitive_of_x_squared_over_a_x_+_b/Proof_2 | [
"Primitive of x squared over a x + b",
"Primitives involving a x + b"
] | [] | [
"Primitive of Power of x by Power of a x + b/Decrement of Power of x",
"Primitive of x over a x + b",
"Definition:Primitive (Calculus)/Constant of Integration"
] |
proofwiki-9134 | Primitive of x cubed over a x + b | :$\ds \int \frac {x^3 \rd x} {a x + b} = \frac {\paren {a x + b}^3} {3 a^4} - \frac {3 b \paren {a x + b}^2} {2 a^4} - \frac {3 b^2 \paren {a x + b} } {a^4} + \frac {b^3} {a^4} \ln \size {a x + b} + C$ | Put $u = a x + b$.
Then:
{{begin-eqn}}
{{eqn | l = x
| r = \frac {u - b} a
| c =
}}
{{eqn | l = \frac {\d u} {\d x}
| r = \frac 1 a
| c =
}}
{{end-eqn}}
Then:
{{begin-eqn}}
{{eqn | l = \int \frac {x^3 \rd x} {a x + b}
| r = \int \frac 1 a \paren {\frac {u - b} a}^3 \frac {\d u} u
|... | :$\ds \int \frac {x^3 \rd x} {a x + b} = \frac {\paren {a x + b}^3} {3 a^4} - \frac {3 b \paren {a x + b}^2} {2 a^4} - \frac {3 b^2 \paren {a x + b} } {a^4} + \frac {b^3} {a^4} \ln \size {a x + b} + C$ | Put $u = a x + b$.
Then:
{{begin-eqn}}
{{eqn | l = x
| r = \frac {u - b} a
| c =
}}
{{eqn | l = \frac {\d u} {\d x}
| r = \frac 1 a
| c =
}}
{{end-eqn}}
Then:
{{begin-eqn}}
{{eqn | l = \int \frac {x^3 \rd x} {a x + b}
| r = \int \frac 1 a \paren {\frac {u - b} a}^3 \frac {\d u} u
... | Primitive of x cubed over a x + b | https://proofwiki.org/wiki/Primitive_of_x_cubed_over_a_x_+_b | https://proofwiki.org/wiki/Primitive_of_x_cubed_over_a_x_+_b | [
"Primitives involving a x + b"
] | [] | [
"Integration by Substitution",
"Primitive of Constant Multiple of Function",
"Linear Combination of Integrals/Indefinite",
"Primitive of Power",
"Primitive of Constant",
"Definition:Primitive (Calculus)/Constant of Integration",
"Primitive of Constant Multiple of Function",
"Primitive of Reciprocal"
] |
proofwiki-9135 | Primitive of Reciprocal of x by a x + b | :$\ds \int \frac {\rd x} {x \paren {a x + b} } = \frac 1 b \ln \size {\frac x {a x + b} } + C$ | {{begin-eqn}}
{{eqn | l = \int \frac {\d x} {x \paren {a x + b}^2}
| r = \int \paren {\frac 1 {b^2 x} - \frac a {b^2 \paren {a x + b} } - \frac a {b \paren {a x + b}^2} } \rd x
| c = Partial Fraction Expansion
}}
{{eqn | r = \frac 1 {b^2} \int \frac {\d x} x - \frac a {b^2} \int \frac {\d x} {a x + b} - \fr... | :$\ds \int \frac {\rd x} {x \paren {a x + b} } = \frac 1 b \ln \size {\frac x {a x + b} } + C$ | {{begin-eqn}}
{{eqn | l = \int \frac {\d x} {x \paren {a x + b}^2}
| r = \int \paren {\frac 1 {b^2 x} - \frac a {b^2 \paren {a x + b} } - \frac a {b \paren {a x + b}^2} } \rd x
| c = [[Primitive of Reciprocal of x by a x + b squared/Partial Fraction Expansion|Partial Fraction Expansion]]
}}
{{eqn | r = \fra... | Primitive of Reciprocal of x by a x + b squared/Proof 1 | https://proofwiki.org/wiki/Primitive_of_Reciprocal_of_x_by_a_x_+_b | https://proofwiki.org/wiki/Primitive_of_Reciprocal_of_x_by_a_x_+_b_squared/Proof_1 | [
"Primitive of Reciprocal of x by a x + b",
"Primitives involving a x + b",
"Primitives involving Reciprocals"
] | [] | [
"Primitive of Reciprocal of x by a x + b squared/Partial Fraction Expansion",
"Linear Combination of Integrals/Indefinite",
"Primitive of Reciprocal",
"Primitive of Reciprocal of a x + b",
"Primitive of Reciprocal of a x + b squared",
"Difference of Logarithms"
] |
proofwiki-9136 | Primitive of Reciprocal of x by a x + b | :$\ds \int \frac {\rd x} {x \paren {a x + b} } = \frac 1 b \ln \size {\frac x {a x + b} } + C$ | {{begin-eqn}}
{{eqn | l = \int \frac {\d x} {x \paren {a x + b}^2}
| r = \int \frac {b \rd x} {b x \paren {a x + b}^2}
| c = multiplying top and bottom by $b$
}}
{{eqn | r = \int \frac {\paren {a x + b - a x} \rd x} {b x \paren {a x + b}^2}
| c = adding and subtracting $a x$
}}
{{eqn | r = \frac 1 b \... | :$\ds \int \frac {\rd x} {x \paren {a x + b} } = \frac 1 b \ln \size {\frac x {a x + b} } + C$ | {{begin-eqn}}
{{eqn | l = \int \frac {\d x} {x \paren {a x + b}^2}
| r = \int \frac {b \rd x} {b x \paren {a x + b}^2}
| c = multiplying [[Definition:Numerator|top]] and [[Definition:Denominator|bottom]] by $b$
}}
{{eqn | r = \int \frac {\paren {a x + b - a x} \rd x} {b x \paren {a x + b}^2}
| c = add... | Primitive of Reciprocal of x by a x + b squared/Proof 2 | https://proofwiki.org/wiki/Primitive_of_Reciprocal_of_x_by_a_x_+_b | https://proofwiki.org/wiki/Primitive_of_Reciprocal_of_x_by_a_x_+_b_squared/Proof_2 | [
"Primitive of Reciprocal of x by a x + b",
"Primitives involving a x + b",
"Primitives involving Reciprocals"
] | [] | [
"Definition:Fraction/Numerator",
"Definition:Fraction/Denominator",
"Linear Combination of Integrals/Indefinite",
"Primitive of Reciprocal of x by a x + b",
"Primitive of Reciprocal of a x + b squared"
] |
proofwiki-9137 | Primitive of Reciprocal of x by a x + b | :$\ds \int \frac {\rd x} {x \paren {a x + b} } = \frac 1 b \ln \size {\frac x {a x + b} } + C$ | {{begin-eqn}}
{{eqn | l = \int \frac {\d x} {x \paren {a x + b} }
| r = \int \paren {\dfrac 1 {b x} - \dfrac a {b \paren {a x + b} } } \rd x
| c = Partial Fraction Expansion
}}
{{eqn | r = \frac 1 b \int \frac {\d x} x - \frac a b \int \frac {\d x} {a x + b}
| c = Linear Combination of Primitives
}}
{... | :$\ds \int \frac {\rd x} {x \paren {a x + b} } = \frac 1 b \ln \size {\frac x {a x + b} } + C$ | {{begin-eqn}}
{{eqn | l = \int \frac {\d x} {x \paren {a x + b} }
| r = \int \paren {\dfrac 1 {b x} - \dfrac a {b \paren {a x + b} } } \rd x
| c = [[Primitive of Reciprocal of x by a x + b/Partial Fraction Expansion|Partial Fraction Expansion]]
}}
{{eqn | r = \frac 1 b \int \frac {\d x} x - \frac a b \int \... | Primitive of Reciprocal of x by a x + b/Proof 1 | https://proofwiki.org/wiki/Primitive_of_Reciprocal_of_x_by_a_x_+_b | https://proofwiki.org/wiki/Primitive_of_Reciprocal_of_x_by_a_x_+_b/Proof_1 | [
"Primitive of Reciprocal of x by a x + b",
"Primitives involving a x + b",
"Primitives involving Reciprocals"
] | [] | [
"Primitive of Reciprocal of x by a x + b/Partial Fraction Expansion",
"Linear Combination of Integrals/Indefinite",
"Primitive of Reciprocal",
"Primitive of Reciprocal of a x + b",
"Difference of Logarithms"
] |
proofwiki-9138 | Primitive of Reciprocal of x by a x + b | :$\ds \int \frac {\rd x} {x \paren {a x + b} } = \frac 1 b \ln \size {\frac x {a x + b} } + C$ | {{begin-eqn}}
{{eqn | l = \int \frac {\d x} {x \paren {a x + b} }
| r = \int \frac {b \rd x} {b x \paren {a x + b} }
| c = multiplying top and bottom by $b$
}}
{{eqn | r = \int \frac {\paren {a x + b - a x} \rd x} {b x \paren {a x + b} }
| c = adding and subtracting $a x$
}}
{{eqn | r = \frac 1 b \int... | :$\ds \int \frac {\rd x} {x \paren {a x + b} } = \frac 1 b \ln \size {\frac x {a x + b} } + C$ | {{begin-eqn}}
{{eqn | l = \int \frac {\d x} {x \paren {a x + b} }
| r = \int \frac {b \rd x} {b x \paren {a x + b} }
| c = multiplying [[Definition:Numerator|top]] and [[Definition:Denominator|bottom]] by $b$
}}
{{eqn | r = \int \frac {\paren {a x + b - a x} \rd x} {b x \paren {a x + b} }
| c = adding... | Primitive of Reciprocal of x by a x + b/Proof 2 | https://proofwiki.org/wiki/Primitive_of_Reciprocal_of_x_by_a_x_+_b | https://proofwiki.org/wiki/Primitive_of_Reciprocal_of_x_by_a_x_+_b/Proof_2 | [
"Primitive of Reciprocal of x by a x + b",
"Primitives involving a x + b",
"Primitives involving Reciprocals"
] | [] | [
"Definition:Fraction/Numerator",
"Definition:Fraction/Denominator",
"Linear Combination of Integrals/Indefinite",
"Primitive of Reciprocal",
"Primitive of Reciprocal of a x + b",
"Difference of Logarithms"
] |
proofwiki-9139 | Primitive of Reciprocal of x squared by a x + b | :$\ds \int \frac {\d x} {x^2 \paren {a x + b} } = -\frac 1 {b x} + \frac a {b^2} \ln \size {\frac {a x + b} x} + C$ | {{begin-eqn}}
{{eqn | l = \int \frac {\d x} {x^2 \paren {a x + b} }
| r = \int \paren {-\frac a {b^2 x} + \frac 1 {b x^2} + \frac {a^2} {b^2 \paren {a x + b} } } \rd x
| c = Partial Fraction Expansion
}}
{{eqn | r = -\frac a {b^2} \int \frac {\d x} x + \frac 1 b \int \frac {\d x} {x^2} + \frac {a^2} {b^2} \... | :$\ds \int \frac {\d x} {x^2 \paren {a x + b} } = -\frac 1 {b x} + \frac a {b^2} \ln \size {\frac {a x + b} x} + C$ | {{begin-eqn}}
{{eqn | l = \int \frac {\d x} {x^2 \paren {a x + b} }
| r = \int \paren {-\frac a {b^2 x} + \frac 1 {b x^2} + \frac {a^2} {b^2 \paren {a x + b} } } \rd x
| c = [[Primitive of Reciprocal of x squared by a x + b/Partial Fraction Expansion|Partial Fraction Expansion]]
}}
{{eqn | r = -\frac a {b^2... | Primitive of Reciprocal of x squared by a x + b/Proof 1 | https://proofwiki.org/wiki/Primitive_of_Reciprocal_of_x_squared_by_a_x_+_b | https://proofwiki.org/wiki/Primitive_of_Reciprocal_of_x_squared_by_a_x_+_b/Proof_1 | [
"Primitive of Reciprocal of x squared by a x + b",
"Primitives involving a x + b",
"Primitives involving Reciprocals"
] | [] | [
"Primitive of Reciprocal of x squared by a x + b/Partial Fraction Expansion",
"Linear Combination of Integrals/Indefinite",
"Primitive of Power",
"Primitive of Reciprocal",
"Primitive of Reciprocal of a x + b",
"Difference of Logarithms"
] |
proofwiki-9140 | Primitive of Reciprocal of x squared by a x + b | :$\ds \int \frac {\d x} {x^2 \paren {a x + b} } = -\frac 1 {b x} + \frac a {b^2} \ln \size {\frac {a x + b} x} + C$ | {{begin-eqn}}
{{eqn | l = \int \frac {\d x} {x^2 \paren {a x + b} }
| r = \int \frac {b \rd x} {b x^2 \paren {a x + b} }
| c = multiplying top and bottom by $b$
}}
{{eqn | r = \int \frac {\paren {a x + b - a x} \rd x} {b x^2 \paren {a x + b} }
| c = adding and subtracting $a x$
}}
{{eqn | r = \frac 1 ... | :$\ds \int \frac {\d x} {x^2 \paren {a x + b} } = -\frac 1 {b x} + \frac a {b^2} \ln \size {\frac {a x + b} x} + C$ | {{begin-eqn}}
{{eqn | l = \int \frac {\d x} {x^2 \paren {a x + b} }
| r = \int \frac {b \rd x} {b x^2 \paren {a x + b} }
| c = multiplying [[Definition:Numerator|top]] and [[Definition:Denominator|bottom]] by $b$
}}
{{eqn | r = \int \frac {\paren {a x + b - a x} \rd x} {b x^2 \paren {a x + b} }
| c = ... | Primitive of Reciprocal of x squared by a x + b/Proof 2 | https://proofwiki.org/wiki/Primitive_of_Reciprocal_of_x_squared_by_a_x_+_b | https://proofwiki.org/wiki/Primitive_of_Reciprocal_of_x_squared_by_a_x_+_b/Proof_2 | [
"Primitive of Reciprocal of x squared by a x + b",
"Primitives involving a x + b",
"Primitives involving Reciprocals"
] | [] | [
"Definition:Fraction/Numerator",
"Definition:Fraction/Denominator",
"Linear Combination of Integrals/Indefinite",
"Primitive of Power",
"Primitive of Reciprocal of x by a x + b",
"Logarithm of Reciprocal"
] |
proofwiki-9141 | Primitive of Reciprocal of x cubed by a x + b | :$\ds \int \frac {\d x} {x^3 \paren {a x + b} } = \frac {2 a x - b} {2 b^2 x^2} + \frac {a^2} {b^3} \ln \size {\frac x {a x + b} } + C$ | {{begin-eqn}}
{{eqn | l = \int \frac {\d x} {x^3 \paren {a x + b} }
| r = \int \paren {\frac {a^2} {b^3 x} + \frac {-a} {b^2 x^2} + \frac 1 {b x^3} + \frac {-a^3} {b^3 \paren {a x + b} } } \rd x
| c = Partial Fraction Expansion
}}
{{eqn | r = \frac {a^2} {b^3} \int \frac {\d x} x + \frac {-a} {b^2} \int \fr... | :$\ds \int \frac {\d x} {x^3 \paren {a x + b} } = \frac {2 a x - b} {2 b^2 x^2} + \frac {a^2} {b^3} \ln \size {\frac x {a x + b} } + C$ | {{begin-eqn}}
{{eqn | l = \int \frac {\d x} {x^3 \paren {a x + b} }
| r = \int \paren {\frac {a^2} {b^3 x} + \frac {-a} {b^2 x^2} + \frac 1 {b x^3} + \frac {-a^3} {b^3 \paren {a x + b} } } \rd x
| c = [[Primitive of Reciprocal of x cubed by a x + b/Partial Fraction Expansion|Partial Fraction Expansion]]
}}
... | Primitive of Reciprocal of x cubed by a x + b/Proof 1 | https://proofwiki.org/wiki/Primitive_of_Reciprocal_of_x_cubed_by_a_x_+_b | https://proofwiki.org/wiki/Primitive_of_Reciprocal_of_x_cubed_by_a_x_+_b/Proof_1 | [
"Primitive of Reciprocal of x cubed by a x + b",
"Primitives involving a x + b",
"Primitives involving Reciprocals"
] | [] | [
"Primitive of Reciprocal of x cubed by a x + b/Partial Fraction Expansion",
"Linear Combination of Integrals/Indefinite",
"Primitive of Power",
"Primitive of Reciprocal",
"Primitive of Reciprocal of a x + b",
"Difference of Logarithms"
] |
proofwiki-9142 | Primitive of Reciprocal of x cubed by a x + b | :$\ds \int \frac {\d x} {x^3 \paren {a x + b} } = \frac {2 a x - b} {2 b^2 x^2} + \frac {a^2} {b^3} \ln \size {\frac x {a x + b} } + C$ | {{begin-eqn}}
{{eqn | l = \int \frac {\d x} {x^3 \paren {a x + b} }
| r = \int \frac {b \rd x} {b x^3 \paren {a x + b} }
| c = multiplying top and bottom by $b$
}}
{{eqn | r = \int \frac {\paren {a x + b - a x} \rd x} {b x^3 \paren {a x + b} }
| c = adding and subtracting $a x$
}}
{{eqn | r = \frac 1 ... | :$\ds \int \frac {\d x} {x^3 \paren {a x + b} } = \frac {2 a x - b} {2 b^2 x^2} + \frac {a^2} {b^3} \ln \size {\frac x {a x + b} } + C$ | {{begin-eqn}}
{{eqn | l = \int \frac {\d x} {x^3 \paren {a x + b} }
| r = \int \frac {b \rd x} {b x^3 \paren {a x + b} }
| c = multiplying [[Definition:Numerator|top]] and [[Definition:Denominator|bottom]] by $b$
}}
{{eqn | r = \int \frac {\paren {a x + b - a x} \rd x} {b x^3 \paren {a x + b} }
| c = ... | Primitive of Reciprocal of x cubed by a x + b/Proof 2 | https://proofwiki.org/wiki/Primitive_of_Reciprocal_of_x_cubed_by_a_x_+_b | https://proofwiki.org/wiki/Primitive_of_Reciprocal_of_x_cubed_by_a_x_+_b/Proof_2 | [
"Primitive of Reciprocal of x cubed by a x + b",
"Primitives involving a x + b",
"Primitives involving Reciprocals"
] | [] | [
"Definition:Fraction/Numerator",
"Definition:Fraction/Denominator",
"Linear Combination of Integrals/Indefinite",
"Primitive of Power",
"Primitive of Reciprocal of x squared by a x + b",
"Difference of Logarithms"
] |
proofwiki-9143 | Primitive of Reciprocal of a x + b squared | :$\ds \int \frac {\d x} {\paren {a x + b}^2} = -\frac 1 {a \paren {a x + b} } + C$ | Let $u = a x + b$.
Then:
{{begin-eqn}}
{{eqn | l = \int \frac {\d x} {\paren {a x + b}^2}
| r = \frac 1 a \int \frac {\d u} {u^2}
| c = Primitive of Function of $a x + b$
}}
{{eqn | r = \frac 1 a \frac {-1} u + C
| c = Primitive of Power
}}
{{eqn | r = -\frac 1 {a \paren {a x + b} } + C
| c = su... | :$\ds \int \frac {\d x} {\paren {a x + b}^2} = -\frac 1 {a \paren {a x + b} } + C$ | Let $u = a x + b$.
Then:
{{begin-eqn}}
{{eqn | l = \int \frac {\d x} {\paren {a x + b}^2}
| r = \frac 1 a \int \frac {\d u} {u^2}
| c = [[Primitive of Function of a x + b|Primitive of Function of $a x + b$]]
}}
{{eqn | r = \frac 1 a \frac {-1} u + C
| c = [[Primitive of Power]]
}}
{{eqn | r = -\frac... | Primitive of Reciprocal of a x + b squared/Proof 1 | https://proofwiki.org/wiki/Primitive_of_Reciprocal_of_a_x_+_b_squared | https://proofwiki.org/wiki/Primitive_of_Reciprocal_of_a_x_+_b_squared/Proof_1 | [
"Primitive of Reciprocal of a x + b squared",
"Primitives involving a x + b",
"Primitives involving Reciprocals"
] | [] | [
"Primitive of Function of a x + b",
"Primitive of Power"
] |
proofwiki-9144 | Primitive of Reciprocal of a x + b squared | :$\ds \int \frac {\d x} {\paren {a x + b}^2} = -\frac 1 {a \paren {a x + b} } + C$ | From Primitive of Power of $a x + b$:
:$\ds \int \paren {a x + b}^n \rd x = \frac {\paren {a x + b}^{n + 1} } {\paren {n + 1} a} + C$
where $n \ne 1$.
The result follows by setting $n = -2$.
{{qed}} | :$\ds \int \frac {\d x} {\paren {a x + b}^2} = -\frac 1 {a \paren {a x + b} } + C$ | From [[Primitive of Power of a x + b|Primitive of Power of $a x + b$]]:
:$\ds \int \paren {a x + b}^n \rd x = \frac {\paren {a x + b}^{n + 1} } {\paren {n + 1} a} + C$
where $n \ne 1$.
The result follows by setting $n = -2$.
{{qed}} | Primitive of Reciprocal of a x + b squared/Proof 2 | https://proofwiki.org/wiki/Primitive_of_Reciprocal_of_a_x_+_b_squared | https://proofwiki.org/wiki/Primitive_of_Reciprocal_of_a_x_+_b_squared/Proof_2 | [
"Primitive of Reciprocal of a x + b squared",
"Primitives involving a x + b",
"Primitives involving Reciprocals"
] | [] | [
"Primitive of Power of a x + b"
] |
proofwiki-9145 | Primitive of x over a x + b squared | :$\ds \int \frac {x \rd x} {\paren {a x + b}^2} = \frac b {a^2 \paren {a x + b} } + \frac 1 {a^2} \ln \size {a x + b} + C$ | Put $u = a x + b$.
Then:
{{begin-eqn}}
{{eqn | l = x
| r = \frac {u - b} a
| c =
}}
{{eqn | l = \frac {\d u} {\d x}
| r = \frac 1 a
| c =
}}
{{end-eqn}}
Then:
{{begin-eqn}}
{{eqn | l = \int \frac {x \rd x} {\paren {a x + b}^2}
| r = \int \frac 1 a \frac {u - b} {a u^2} \rd u
| c = ... | :$\ds \int \frac {x \rd x} {\paren {a x + b}^2} = \frac b {a^2 \paren {a x + b} } + \frac 1 {a^2} \ln \size {a x + b} + C$ | Put $u = a x + b$.
Then:
{{begin-eqn}}
{{eqn | l = x
| r = \frac {u - b} a
| c =
}}
{{eqn | l = \frac {\d u} {\d x}
| r = \frac 1 a
| c =
}}
{{end-eqn}}
Then:
{{begin-eqn}}
{{eqn | l = \int \frac {x \rd x} {\paren {a x + b}^2}
| r = \int \frac 1 a \frac {u - b} {a u^2} \rd u
| ... | Primitive of x over a x + b squared/Proof 1 | https://proofwiki.org/wiki/Primitive_of_x_over_a_x_+_b_squared | https://proofwiki.org/wiki/Primitive_of_x_over_a_x_+_b_squared/Proof_1 | [
"Primitive of x over a x + b squared",
"Primitives involving a x + b"
] | [] | [
"Integration by Substitution",
"Linear Combination of Integrals/Indefinite",
"Primitive of Reciprocal",
"Primitive of Power"
] |
proofwiki-9146 | Primitive of x over a x + b squared | :$\ds \int \frac {x \rd x} {\paren {a x + b}^2} = \frac b {a^2 \paren {a x + b} } + \frac 1 {a^2} \ln \size {a x + b} + C$ | {{begin-eqn}}
{{eqn | l = \int \frac {x \rd x} {\paren {a x + b}^2}
| r = \int \frac {a x \rd x} {a \paren {a x + b}^2}
| c = multiplying top and bottom by $a$
}}
{{eqn | r = \int \frac {\paren {a x + b - b} \rd x} {a \paren {a x + b}^2}
| c = adding and subtracting $b$
}}
{{eqn | r = \frac 1 a \int \... | :$\ds \int \frac {x \rd x} {\paren {a x + b}^2} = \frac b {a^2 \paren {a x + b} } + \frac 1 {a^2} \ln \size {a x + b} + C$ | {{begin-eqn}}
{{eqn | l = \int \frac {x \rd x} {\paren {a x + b}^2}
| r = \int \frac {a x \rd x} {a \paren {a x + b}^2}
| c = multiplying [[Definition:Numerator|top]] and [[Definition:Denominator|bottom]] by $a$
}}
{{eqn | r = \int \frac {\paren {a x + b - b} \rd x} {a \paren {a x + b}^2}
| c = adding... | Primitive of x over a x + b squared/Proof 2 | https://proofwiki.org/wiki/Primitive_of_x_over_a_x_+_b_squared | https://proofwiki.org/wiki/Primitive_of_x_over_a_x_+_b_squared/Proof_2 | [
"Primitive of x over a x + b squared",
"Primitives involving a x + b"
] | [] | [
"Definition:Fraction/Numerator",
"Definition:Fraction/Denominator",
"Linear Combination of Integrals/Indefinite",
"Primitive of Reciprocal of a x + b",
"Primitive of Reciprocal of a x + b squared"
] |
proofwiki-9147 | Primitive of x squared over a x + b squared | :$\ds \int \frac {x^2 \rd x} {\paren {a x + b}^2} = \frac {a x + b} {a^3} - \frac {b^2} {a^3 \paren {a x + b} } - \frac {2 b} {a^3} \ln \size {a x + b} + C$ | Put $u = a x + b$.
Then:
{{begin-eqn}}
{{eqn | l = x
| r = \frac {u - b} a
| c =
}}
{{eqn | l = \frac {\d u} {\d x}
| r = \frac 1 a
| c =
}}
{{end-eqn}}
Then:
{{begin-eqn}}
{{eqn | l = \int \frac {x^2 \rd x} {\paren {a x + b}^2}
| r = \int \frac 1 a \paren {\frac {u - b} a}^2 \frac 1 {u^... | :$\ds \int \frac {x^2 \rd x} {\paren {a x + b}^2} = \frac {a x + b} {a^3} - \frac {b^2} {a^3 \paren {a x + b} } - \frac {2 b} {a^3} \ln \size {a x + b} + C$ | Put $u = a x + b$.
Then:
{{begin-eqn}}
{{eqn | l = x
| r = \frac {u - b} a
| c =
}}
{{eqn | l = \frac {\d u} {\d x}
| r = \frac 1 a
| c =
}}
{{end-eqn}}
Then:
{{begin-eqn}}
{{eqn | l = \int \frac {x^2 \rd x} {\paren {a x + b}^2}
| r = \int \frac 1 a \paren {\frac {u - b} a}^2 \frac 1... | Primitive of x squared over a x + b squared/Proof 1 | https://proofwiki.org/wiki/Primitive_of_x_squared_over_a_x_+_b_squared | https://proofwiki.org/wiki/Primitive_of_x_squared_over_a_x_+_b_squared/Proof_1 | [
"Primitive of x squared over a x + b squared",
"Primitives involving a x + b"
] | [] | [
"Integration by Substitution",
"Square of Difference",
"Linear Combination of Integrals/Indefinite",
"Primitive of Constant",
"Primitive of Reciprocal",
"Primitive of Power"
] |
proofwiki-9148 | Primitive of x squared over a x + b squared | :$\ds \int \frac {x^2 \rd x} {\paren {a x + b}^2} = \frac {a x + b} {a^3} - \frac {b^2} {a^3 \paren {a x + b} } - \frac {2 b} {a^3} \ln \size {a x + b} + C$ | {{begin-eqn}}
{{eqn | l = \int \frac {x^2 \rd x} {\paren {a x + b}^2}
| r = \int \frac {a x^2 \rd x} {a \paren {a x + b}^2}
| c = multiplying top and bottom by $a$
}}
{{eqn | r = \int \frac {x \paren {a x + b - b} \rd x} {a \paren {a x + b}^2}
| c = adding and subtracting $b x$
}}
{{eqn | r = \frac 1 ... | :$\ds \int \frac {x^2 \rd x} {\paren {a x + b}^2} = \frac {a x + b} {a^3} - \frac {b^2} {a^3 \paren {a x + b} } - \frac {2 b} {a^3} \ln \size {a x + b} + C$ | {{begin-eqn}}
{{eqn | l = \int \frac {x^2 \rd x} {\paren {a x + b}^2}
| r = \int \frac {a x^2 \rd x} {a \paren {a x + b}^2}
| c = multiplying [[Definition:Numerator|top]] and [[Definition:Denominator|bottom]] by $a$
}}
{{eqn | r = \int \frac {x \paren {a x + b - b} \rd x} {a \paren {a x + b}^2}
| c = ... | Primitive of x squared over a x + b squared/Proof 2 | https://proofwiki.org/wiki/Primitive_of_x_squared_over_a_x_+_b_squared | https://proofwiki.org/wiki/Primitive_of_x_squared_over_a_x_+_b_squared/Proof_2 | [
"Primitive of x squared over a x + b squared",
"Primitives involving a x + b"
] | [] | [
"Definition:Fraction/Numerator",
"Definition:Fraction/Denominator",
"Linear Combination of Integrals/Indefinite",
"Primitive of x over a x + b",
"Primitive of x over a x + b squared"
] |
proofwiki-9149 | Primitive of x cubed over a x + b squared | :$\ds \int \frac {x^3 \rd x} {\paren {a x + b}^2} = \frac {\paren {a x + b}^2} {2 a^4} - \frac {3 b \paren {a x + b} } {a^4} + \frac {b^3} {a^4 \paren {a x + b} } + \frac {3 b^2} {a^4} \ln \size {a x + b} + C$ | Put $u = a x + b$.
Then:
{{begin-eqn}}
{{eqn | l = x
| r = \frac {u - b} a
| c =
}}
{{eqn | l = \frac {\d u} {\d x}
| r = \frac 1 a
| c =
}}
{{end-eqn}}
Then:
{{begin-eqn}}
{{eqn | l = \int \frac {x^3 \rd x} {\paren {a x + b}^2}
| r = \int \frac 1 a \paren {\frac {u - b} a}^3 \frac 1 {u... | :$\ds \int \frac {x^3 \rd x} {\paren {a x + b}^2} = \frac {\paren {a x + b}^2} {2 a^4} - \frac {3 b \paren {a x + b} } {a^4} + \frac {b^3} {a^4 \paren {a x + b} } + \frac {3 b^2} {a^4} \ln \size {a x + b} + C$ | Put $u = a x + b$.
Then:
{{begin-eqn}}
{{eqn | l = x
| r = \frac {u - b} a
| c =
}}
{{eqn | l = \frac {\d u} {\d x}
| r = \frac 1 a
| c =
}}
{{end-eqn}}
Then:
{{begin-eqn}}
{{eqn | l = \int \frac {x^3 \rd x} {\paren {a x + b}^2}
| r = \int \frac 1 a \paren {\frac {u - b} a}^3 \frac ... | Primitive of x cubed over a x + b squared | https://proofwiki.org/wiki/Primitive_of_x_cubed_over_a_x_+_b_squared | https://proofwiki.org/wiki/Primitive_of_x_cubed_over_a_x_+_b_squared | [
"Primitives involving a x + b"
] | [] | [
"Integration by Substitution",
"Linear Combination of Integrals/Indefinite",
"Primitive of Power",
"Primitive of Constant",
"Primitive of Reciprocal"
] |
proofwiki-9150 | Primitive of Reciprocal of x by a x + b squared | :$\ds \int \frac {\d x} {x \paren {a x + b}^2} = \frac 1 {b \paren {a x + b} } + \frac 1 {b^2} \ln \size {\frac x {a x + b} } + C$ | {{begin-eqn}}
{{eqn | l = \int \frac {\d x} {x \paren {a x + b}^2}
| r = \int \paren {\frac 1 {b^2 x} - \frac a {b^2 \paren {a x + b} } - \frac a {b \paren {a x + b}^2} } \rd x
| c = Partial Fraction Expansion
}}
{{eqn | r = \frac 1 {b^2} \int \frac {\d x} x - \frac a {b^2} \int \frac {\d x} {a x + b} - \fr... | :$\ds \int \frac {\d x} {x \paren {a x + b}^2} = \frac 1 {b \paren {a x + b} } + \frac 1 {b^2} \ln \size {\frac x {a x + b} } + C$ | {{begin-eqn}}
{{eqn | l = \int \frac {\d x} {x \paren {a x + b}^2}
| r = \int \paren {\frac 1 {b^2 x} - \frac a {b^2 \paren {a x + b} } - \frac a {b \paren {a x + b}^2} } \rd x
| c = [[Primitive of Reciprocal of x by a x + b squared/Partial Fraction Expansion|Partial Fraction Expansion]]
}}
{{eqn | r = \fra... | Primitive of Reciprocal of x by a x + b squared/Proof 1 | https://proofwiki.org/wiki/Primitive_of_Reciprocal_of_x_by_a_x_+_b_squared | https://proofwiki.org/wiki/Primitive_of_Reciprocal_of_x_by_a_x_+_b_squared/Proof_1 | [
"Primitive of Reciprocal of x by a x + b squared",
"Primitives involving a x + b",
"Primitives involving Reciprocals"
] | [] | [
"Primitive of Reciprocal of x by a x + b squared/Partial Fraction Expansion",
"Linear Combination of Integrals/Indefinite",
"Primitive of Reciprocal",
"Primitive of Reciprocal of a x + b",
"Primitive of Reciprocal of a x + b squared",
"Difference of Logarithms"
] |
proofwiki-9151 | Primitive of Reciprocal of x by a x + b squared | :$\ds \int \frac {\d x} {x \paren {a x + b}^2} = \frac 1 {b \paren {a x + b} } + \frac 1 {b^2} \ln \size {\frac x {a x + b} } + C$ | {{begin-eqn}}
{{eqn | l = \int \frac {\d x} {x \paren {a x + b}^2}
| r = \int \frac {b \rd x} {b x \paren {a x + b}^2}
| c = multiplying top and bottom by $b$
}}
{{eqn | r = \int \frac {\paren {a x + b - a x} \rd x} {b x \paren {a x + b}^2}
| c = adding and subtracting $a x$
}}
{{eqn | r = \frac 1 b \... | :$\ds \int \frac {\d x} {x \paren {a x + b}^2} = \frac 1 {b \paren {a x + b} } + \frac 1 {b^2} \ln \size {\frac x {a x + b} } + C$ | {{begin-eqn}}
{{eqn | l = \int \frac {\d x} {x \paren {a x + b}^2}
| r = \int \frac {b \rd x} {b x \paren {a x + b}^2}
| c = multiplying [[Definition:Numerator|top]] and [[Definition:Denominator|bottom]] by $b$
}}
{{eqn | r = \int \frac {\paren {a x + b - a x} \rd x} {b x \paren {a x + b}^2}
| c = add... | Primitive of Reciprocal of x by a x + b squared/Proof 2 | https://proofwiki.org/wiki/Primitive_of_Reciprocal_of_x_by_a_x_+_b_squared | https://proofwiki.org/wiki/Primitive_of_Reciprocal_of_x_by_a_x_+_b_squared/Proof_2 | [
"Primitive of Reciprocal of x by a x + b squared",
"Primitives involving a x + b",
"Primitives involving Reciprocals"
] | [] | [
"Definition:Fraction/Numerator",
"Definition:Fraction/Denominator",
"Linear Combination of Integrals/Indefinite",
"Primitive of Reciprocal of x by a x + b",
"Primitive of Reciprocal of a x + b squared"
] |
proofwiki-9152 | Primitive of Reciprocal of x squared by a x + b squared | :$\ds \int \frac {\d x} {x^2 \paren {a x + b}^2} = \frac {-a} {b^2 \paren {a x + b} } - \frac 1 {b^2 x} + \frac {2 a} {b^3} \ln \size {\frac {a x + b} x} + C$ | {{begin-eqn}}
{{eqn | l = \int \frac {\d x} {x^2 \paren {a x + b}^2}
| r = \int \paren {-\frac {2 a} {b^3 x} + \frac 1 {b^2 x^2} + \frac {2 a^2} {b^3 \paren {a x + b} } + \frac {a^2} {b^2 \paren {a x + b}^2} } \rd x
| c = Partial Fraction Expansion
}}
{{eqn | r = -\frac {2 a} {b^3} \int \frac {\d x} x + \fr... | :$\ds \int \frac {\d x} {x^2 \paren {a x + b}^2} = \frac {-a} {b^2 \paren {a x + b} } - \frac 1 {b^2 x} + \frac {2 a} {b^3} \ln \size {\frac {a x + b} x} + C$ | {{begin-eqn}}
{{eqn | l = \int \frac {\d x} {x^2 \paren {a x + b}^2}
| r = \int \paren {-\frac {2 a} {b^3 x} + \frac 1 {b^2 x^2} + \frac {2 a^2} {b^3 \paren {a x + b} } + \frac {a^2} {b^2 \paren {a x + b}^2} } \rd x
| c = [[Primitive of Reciprocal of x squared by a x + b squared/Partial Fraction Expansion|P... | Primitive of Reciprocal of x squared by a x + b squared | https://proofwiki.org/wiki/Primitive_of_Reciprocal_of_x_squared_by_a_x_+_b_squared | https://proofwiki.org/wiki/Primitive_of_Reciprocal_of_x_squared_by_a_x_+_b_squared | [
"Primitives involving a x + b",
"Primitives involving Reciprocals"
] | [] | [
"Primitive of Reciprocal of x squared by a x + b squared/Partial Fraction Expansion",
"Linear Combination of Integrals/Indefinite",
"Primitive of Power",
"Primitive of Reciprocal",
"Primitive of Reciprocal of a x + b",
"Primitive of Reciprocal of a x + b squared",
"Difference of Logarithms"
] |
proofwiki-9153 | Primitive of Reciprocal of x cubed by a x + b squared | :$\ds \int \frac {\d x} {x^3 \paren {a x + b}^2} = - \frac {\paren {a x + b}^2} {2 b^4 x^2} + \frac {3 a \paren {a x + b} } {b^4 x} - \frac {a^3 x} {b^4 \paren {a x + b} } + \frac {3 a^2} {b^4} \ln \size {\frac x {a x + b} } + C$ | {{begin-eqn}}
{{eqn | l = \int \frac {\d x} {x^3 \paren {a x + b}^2}
| r = \int \paren {\frac {3 a^2} {b^4 x} + \frac {-2 a} {b^3 x^2} + \frac 1 {b^2 x^3} + \frac {-3 a^3} {b^4 \paren {a x + b} } + \frac {-a^3} {b^3 \paren {a x + b}^2} } \rd x
| c = Partial Fraction Expansion
}}
{{eqn | r = \frac {3 a^2} {b... | :$\ds \int \frac {\d x} {x^3 \paren {a x + b}^2} = - \frac {\paren {a x + b}^2} {2 b^4 x^2} + \frac {3 a \paren {a x + b} } {b^4 x} - \frac {a^3 x} {b^4 \paren {a x + b} } + \frac {3 a^2} {b^4} \ln \size {\frac x {a x + b} } + C$ | {{begin-eqn}}
{{eqn | l = \int \frac {\d x} {x^3 \paren {a x + b}^2}
| r = \int \paren {\frac {3 a^2} {b^4 x} + \frac {-2 a} {b^3 x^2} + \frac 1 {b^2 x^3} + \frac {-3 a^3} {b^4 \paren {a x + b} } + \frac {-a^3} {b^3 \paren {a x + b}^2} } \rd x
| c = [[Primitive of Reciprocal of x cubed by a x + b squared/Pa... | Primitive of Reciprocal of x cubed by a x + b squared | https://proofwiki.org/wiki/Primitive_of_Reciprocal_of_x_cubed_by_a_x_+_b_squared | https://proofwiki.org/wiki/Primitive_of_Reciprocal_of_x_cubed_by_a_x_+_b_squared | [
"Primitives involving a x + b",
"Primitives involving Reciprocals"
] | [] | [
"Primitive of Reciprocal of x cubed by a x + b squared/Partial Fraction Expansion",
"Linear Combination of Integrals/Indefinite",
"Primitive of Power",
"Primitive of Reciprocal",
"Primitive of Reciprocal of a x + b",
"Primitive of Reciprocal of a x + b squared",
"Difference of Logarithms"
] |
proofwiki-9154 | Primitive of Reciprocal of a x + b cubed | :$\ds \int \frac {\d x} {\paren {a x + b}^3} = \frac {-1} {2 a \paren {a x + b}^2} + C$ | Let $u = a x + b$.
Then:
{{begin-eqn}}
{{eqn | l = \int \frac {\rd x} {\paren {a x + b}^3}
| r = \frac 1 a \int \frac {\rd u} {u^3}
| c = Primitive of Function of $a x + b$
}}
{{eqn | r = \frac 1 a \frac {-1} {2 u^2} + C
| c = Primitive of Power
}}
{{eqn | r = \frac {-1} {2 a \paren {a x + b}^2} + C
... | :$\ds \int \frac {\d x} {\paren {a x + b}^3} = \frac {-1} {2 a \paren {a x + b}^2} + C$ | Let $u = a x + b$.
Then:
{{begin-eqn}}
{{eqn | l = \int \frac {\rd x} {\paren {a x + b}^3}
| r = \frac 1 a \int \frac {\rd u} {u^3}
| c = [[Primitive of Function of a x + b|Primitive of Function of $a x + b$]]
}}
{{eqn | r = \frac 1 a \frac {-1} {2 u^2} + C
| c = [[Primitive of Power]]
}}
{{eqn | r ... | Primitive of Reciprocal of a x + b cubed/Proof 1 | https://proofwiki.org/wiki/Primitive_of_Reciprocal_of_a_x_+_b_cubed | https://proofwiki.org/wiki/Primitive_of_Reciprocal_of_a_x_+_b_cubed/Proof_1 | [
"Primitive of Reciprocal of a x + b cubed",
"Primitives involving a x + b",
"Primitives involving Reciprocals"
] | [] | [
"Primitive of Function of a x + b",
"Primitive of Power"
] |
proofwiki-9155 | Primitive of Reciprocal of a x + b cubed | :$\ds \int \frac {\d x} {\paren {a x + b}^3} = \frac {-1} {2 a \paren {a x + b}^2} + C$ | From Primitive of Power of $a x + b$:
:$\ds \int \paren {a x + b}^n \rd x = \frac {\paren {a x + b}^{n + 1} } {\paren {n + 1} a} + C$
where $n \ne 1$.
The result follows by setting $n = -3$.
{{qed}} | :$\ds \int \frac {\d x} {\paren {a x + b}^3} = \frac {-1} {2 a \paren {a x + b}^2} + C$ | From [[Primitive of Power of a x + b|Primitive of Power of $a x + b$]]:
:$\ds \int \paren {a x + b}^n \rd x = \frac {\paren {a x + b}^{n + 1} } {\paren {n + 1} a} + C$
where $n \ne 1$.
The result follows by setting $n = -3$.
{{qed}} | Primitive of Reciprocal of a x + b cubed/Proof 2 | https://proofwiki.org/wiki/Primitive_of_Reciprocal_of_a_x_+_b_cubed | https://proofwiki.org/wiki/Primitive_of_Reciprocal_of_a_x_+_b_cubed/Proof_2 | [
"Primitive of Reciprocal of a x + b cubed",
"Primitives involving a x + b",
"Primitives involving Reciprocals"
] | [] | [
"Primitive of Power of a x + b"
] |
proofwiki-9156 | Primitive of x over a x + b cubed | :$\ds \int \frac {x \rd x} {\paren {a x + b}^3} = \frac {-1} {a^2 \paren {a x + b} } + \frac b {2 a^2 \paren {a x + b}^2} + C$ | Put $u = a x + b$.
Then:
{{begin-eqn}}
{{eqn | l = x
| r = \frac {u - b} a
| c =
}}
{{eqn | l = \frac {\d u} {\d x}
| r = \frac 1 a
| c =
}}
{{end-eqn}}
Then:
{{begin-eqn}}
{{eqn | l = \int \frac {x \rd x} {\paren {a x + b}^3}
| r = \int \frac 1 a \frac {u - b} {a u^3} \rd u
| c = ... | :$\ds \int \frac {x \rd x} {\paren {a x + b}^3} = \frac {-1} {a^2 \paren {a x + b} } + \frac b {2 a^2 \paren {a x + b}^2} + C$ | Put $u = a x + b$.
Then:
{{begin-eqn}}
{{eqn | l = x
| r = \frac {u - b} a
| c =
}}
{{eqn | l = \frac {\d u} {\d x}
| r = \frac 1 a
| c =
}}
{{end-eqn}}
Then:
{{begin-eqn}}
{{eqn | l = \int \frac {x \rd x} {\paren {a x + b}^3}
| r = \int \frac 1 a \frac {u - b} {a u^3} \rd u
| ... | Primitive of x over a x + b cubed/Proof 1 | https://proofwiki.org/wiki/Primitive_of_x_over_a_x_+_b_cubed | https://proofwiki.org/wiki/Primitive_of_x_over_a_x_+_b_cubed/Proof_1 | [
"Primitive of x over a x + b cubed",
"Primitives involving a x + b"
] | [] | [
"Integration by Substitution",
"Linear Combination of Integrals/Indefinite",
"Primitive of Power"
] |
proofwiki-9157 | Primitive of x over a x + b cubed | :$\ds \int \frac {x \rd x} {\paren {a x + b}^3} = \frac {-1} {a^2 \paren {a x + b} } + \frac b {2 a^2 \paren {a x + b}^2} + C$ | From Primitive of $x$ by Power of $a x + b$:
:$\ds \int x \paren {a x + b}^n \rd x = \frac {\paren {a x + b}^{n + 2} } {\paren {n + 2} a^2} - \frac {b \paren {a x + b}^{n + 1} } {\paren {n + 1} a^2} + C$
where $n \ne - 1$ and $n \ne - 2$.
The result follows by setting $n = -3$.
{{qed}} | :$\ds \int \frac {x \rd x} {\paren {a x + b}^3} = \frac {-1} {a^2 \paren {a x + b} } + \frac b {2 a^2 \paren {a x + b}^2} + C$ | From [[Primitive of x by Power of a x + b|Primitive of $x$ by Power of $a x + b$]]:
:$\ds \int x \paren {a x + b}^n \rd x = \frac {\paren {a x + b}^{n + 2} } {\paren {n + 2} a^2} - \frac {b \paren {a x + b}^{n + 1} } {\paren {n + 1} a^2} + C$
where $n \ne - 1$ and $n \ne - 2$.
The result follows by setting $n = -3$.
... | Primitive of x over a x + b cubed/Proof 2 | https://proofwiki.org/wiki/Primitive_of_x_over_a_x_+_b_cubed | https://proofwiki.org/wiki/Primitive_of_x_over_a_x_+_b_cubed/Proof_2 | [
"Primitive of x over a x + b cubed",
"Primitives involving a x + b"
] | [] | [
"Primitive of x by Power of a x + b"
] |
proofwiki-9158 | Primitive of x squared over a x + b cubed | :$\ds \int \frac {x^2 \rd x} {\paren {a x + b}^3} = \frac {2 b} {a^3 \paren {a x + b} } - \frac {b^2} {2 a^3 \paren {a x + b}^2} + \frac 1 {a^3} \ln \size {a x + b} + C$ | Put $u = a x + b$.
Then:
{{begin-eqn}}
{{eqn | l = x
| r = \frac {u - b} a
| c =
}}
{{eqn | l = \frac {\d u} {\d x}
| r = \frac 1 a
| c =
}}
{{end-eqn}}
Then:
{{begin-eqn}}
{{eqn | l = \int \frac {x^2 \rd x} {\paren {a x + b}^3}
| r = \int \frac 1 a \paren {\frac {u - b} a}^2 \frac 1 {u^... | :$\ds \int \frac {x^2 \rd x} {\paren {a x + b}^3} = \frac {2 b} {a^3 \paren {a x + b} } - \frac {b^2} {2 a^3 \paren {a x + b}^2} + \frac 1 {a^3} \ln \size {a x + b} + C$ | Put $u = a x + b$.
Then:
{{begin-eqn}}
{{eqn | l = x
| r = \frac {u - b} a
| c =
}}
{{eqn | l = \frac {\d u} {\d x}
| r = \frac 1 a
| c =
}}
{{end-eqn}}
Then:
{{begin-eqn}}
{{eqn | l = \int \frac {x^2 \rd x} {\paren {a x + b}^3}
| r = \int \frac 1 a \paren {\frac {u - b} a}^2 \frac 1... | Primitive of x squared over a x + b cubed | https://proofwiki.org/wiki/Primitive_of_x_squared_over_a_x_+_b_cubed | https://proofwiki.org/wiki/Primitive_of_x_squared_over_a_x_+_b_cubed | [
"Primitives involving a x + b"
] | [] | [
"Integration by Substitution",
"Square of Difference",
"Linear Combination of Integrals/Indefinite",
"Primitive of Reciprocal",
"Primitive of Power"
] |
proofwiki-9159 | Primitive of x cubed over a x + b cubed | :$\ds \int \frac {x^3 \rd x} {\paren {a x + b}^3} = \frac x {a^3} - \frac {3 b^2} {a^4 \paren {a x + b} } + \frac {b^3} {2 a^4 \paren {a x + b}^2} - \frac {3 b} {a^4} \ln \size {a x + b} + C$ | Put $u = a x + b$.
Then:
{{begin-eqn}}
{{eqn | l = x
| r = \frac {u - b} a
| c =
}}
{{eqn | l = \frac {\d u} {\d x}
| r = \frac 1 a
| c =
}}
{{end-eqn}}
Then:
{{begin-eqn}}
{{eqn | l = \int \frac {x^3 \rd x} {\paren {a x + b}^3}
| r = \int \frac 1 a \paren {\frac {u - b} a}^3 \frac 1 {u^... | :$\ds \int \frac {x^3 \rd x} {\paren {a x + b}^3} = \frac x {a^3} - \frac {3 b^2} {a^4 \paren {a x + b} } + \frac {b^3} {2 a^4 \paren {a x + b}^2} - \frac {3 b} {a^4} \ln \size {a x + b} + C$ | Put $u = a x + b$.
Then:
{{begin-eqn}}
{{eqn | l = x
| r = \frac {u - b} a
| c =
}}
{{eqn | l = \frac {\d u} {\d x}
| r = \frac 1 a
| c =
}}
{{end-eqn}}
Then:
{{begin-eqn}}
{{eqn | l = \int \frac {x^3 \rd x} {\paren {a x + b}^3}
| r = \int \frac 1 a \paren {\frac {u - b} a}^3 \frac 1... | Primitive of x cubed over a x + b cubed | https://proofwiki.org/wiki/Primitive_of_x_cubed_over_a_x_+_b_cubed | https://proofwiki.org/wiki/Primitive_of_x_cubed_over_a_x_+_b_cubed | [
"Primitives involving a x + b"
] | [] | [
"Integration by Substitution",
"Linear Combination of Integrals/Indefinite",
"Primitive of Constant",
"Primitive of Power",
"Primitive of Reciprocal",
"Definition:Primitive (Calculus)/Constant of Integration"
] |
proofwiki-9160 | Primitive of Reciprocal of x by a x + b cubed | :$\ds\int \frac {\d x} {x \paren {a x + b}^3} = \frac {a^2 x^2} {2 b^3 \paren {a x + b}^2} - \frac {2 a x} {b^3 \paren {a x + b} } + \frac 1 {b^3} \ln \size {\frac x {a x + b} } + C$ | {{begin-eqn}}
{{eqn | l = \int \frac {\d x} {x \paren {a x + b}^3}
| r = \int \paren {\frac 1 {b^3 x} - \frac a {b^3 \paren {a x + b} } - \frac a {b^2 \paren {a x + b}^2} - \frac a {b \paren {a x + b}^3} } \rd x
| c = Partial Fraction Expansion
}}
{{eqn | r = \frac 1 {b^3} \int \frac {\d x} x - \frac a {b^3... | :$\ds\int \frac {\d x} {x \paren {a x + b}^3} = \frac {a^2 x^2} {2 b^3 \paren {a x + b}^2} - \frac {2 a x} {b^3 \paren {a x + b} } + \frac 1 {b^3} \ln \size {\frac x {a x + b} } + C$ | {{begin-eqn}}
{{eqn | l = \int \frac {\d x} {x \paren {a x + b}^3}
| r = \int \paren {\frac 1 {b^3 x} - \frac a {b^3 \paren {a x + b} } - \frac a {b^2 \paren {a x + b}^2} - \frac a {b \paren {a x + b}^3} } \rd x
| c = [[Primitive of Reciprocal of x by a x + b cubed/Partial Fraction Expansion|Partial Fractio... | Primitive of Reciprocal of x by a x + b cubed | https://proofwiki.org/wiki/Primitive_of_Reciprocal_of_x_by_a_x_+_b_cubed | https://proofwiki.org/wiki/Primitive_of_Reciprocal_of_x_by_a_x_+_b_cubed | [
"Primitives involving a x + b"
] | [] | [
"Primitive of Reciprocal of x by a x + b cubed/Partial Fraction Expansion",
"Linear Combination of Integrals/Indefinite",
"Primitive of Reciprocal",
"Primitive of Reciprocal of a x + b",
"Primitive of Reciprocal of a x + b squared",
"Primitive of Reciprocal of a x + b cubed",
"Difference of Logarithms"
... |
proofwiki-9161 | Primitive of Reciprocal of x squared by a x + b cubed | :$\ds \int \frac {\d x} {x^2 \paren {a x + b}^3} = \frac {-a} {2 b^2 \paren {a x + b}^2} - \frac {2 a} {b^3 \paren {a x + b} } - \frac 1 {b^3 x} + \frac {3 a} {b^4} \ln \size {\frac {a x + b} x} + C$ | {{begin-eqn}}
{{eqn | l = \int \frac {\d x} {x^2 \paren {a x + b}^3}
| r = \int \paren {\frac {-3 a} {b^4 x} + \frac 1 {b^3 x^2} + \frac {3 a^2} {b^4 \paren {a x + b} } + \frac {2 a^2} {b^3 \paren {a x + b}\^2} + \frac {a^2} {b^2 \paren {a x + b}^3} } \rd x
| c = Partial Fraction Expansion
}}
{{eqn | r = \f... | :$\ds \int \frac {\d x} {x^2 \paren {a x + b}^3} = \frac {-a} {2 b^2 \paren {a x + b}^2} - \frac {2 a} {b^3 \paren {a x + b} } - \frac 1 {b^3 x} + \frac {3 a} {b^4} \ln \size {\frac {a x + b} x} + C$ | {{begin-eqn}}
{{eqn | l = \int \frac {\d x} {x^2 \paren {a x + b}^3}
| r = \int \paren {\frac {-3 a} {b^4 x} + \frac 1 {b^3 x^2} + \frac {3 a^2} {b^4 \paren {a x + b} } + \frac {2 a^2} {b^3 \paren {a x + b}\^2} + \frac {a^2} {b^2 \paren {a x + b}^3} } \rd x
| c = [[Primitive of Reciprocal of x squared by a ... | Primitive of Reciprocal of x squared by a x + b cubed | https://proofwiki.org/wiki/Primitive_of_Reciprocal_of_x_squared_by_a_x_+_b_cubed | https://proofwiki.org/wiki/Primitive_of_Reciprocal_of_x_squared_by_a_x_+_b_cubed | [
"Primitives involving a x + b",
"Primitives involving Reciprocals"
] | [] | [
"Primitive of Reciprocal of x squared by a x + b cubed/Partial Fraction Expansion",
"Linear Combination of Integrals/Indefinite",
"Primitive of Reciprocal",
"Primitive of Reciprocal of a x + b",
"Primitive of Reciprocal of a x + b squared",
"Primitive of Reciprocal of a x + b cubed",
"Primitive of Power... |
proofwiki-9162 | Primitive of Reciprocal of x cubed by a x + b cubed | :$\ds \int \frac {\d x} {x^3 \paren {a x + b}^3} = \frac {a^4 x^2} {2 b^5 \paren {a x + b}^2} - \frac {4 a^3 x} {b^5 \paren {a x + b} } - \frac {\paren {a x + b}^2} {2 b^5 x^2} + \frac {4 a} {b^4 x} + \frac {6 a^2} {b^5} \ln \size {\frac x {a x + b} } + C$ | A partial fraction expansion of the integrand gives:
:$\dfrac 1 {x^3 \paren {a x + b}^3} = \dfrac {6 a^2} {b^5 x} - \dfrac {3 a} {b^4 x^2} + \dfrac 1 {b^3 x^3} - \dfrac {6 a^3} {b^5 \paren {a x + b} } - \dfrac {3 a^3} {b^4 \paren {a x + b}^2} - \dfrac {a^3} {b^3 \paren {a x + b}^3}$
From Linear Combination of Primitive... | :$\ds \int \frac {\d x} {x^3 \paren {a x + b}^3} = \frac {a^4 x^2} {2 b^5 \paren {a x + b}^2} - \frac {4 a^3 x} {b^5 \paren {a x + b} } - \frac {\paren {a x + b}^2} {2 b^5 x^2} + \frac {4 a} {b^4 x} + \frac {6 a^2} {b^5} \ln \size {\frac x {a x + b} } + C$ | A [[Primitive of Reciprocal of x cubed by a x + b cubed/Partial Fraction Expansion|partial fraction expansion]] of the [[Definition:Integrand|integrand]] gives:
:$\dfrac 1 {x^3 \paren {a x + b}^3} = \dfrac {6 a^2} {b^5 x} - \dfrac {3 a} {b^4 x^2} + \dfrac 1 {b^3 x^3} - \dfrac {6 a^3} {b^5 \paren {a x + b} } - \dfrac {3... | Primitive of Reciprocal of x cubed by a x + b cubed | https://proofwiki.org/wiki/Primitive_of_Reciprocal_of_x_cubed_by_a_x_+_b_cubed | https://proofwiki.org/wiki/Primitive_of_Reciprocal_of_x_cubed_by_a_x_+_b_cubed | [
"Primitives involving a x + b"
] | [] | [
"Primitive of Reciprocal of x cubed by a x + b cubed/Partial Fraction Expansion",
"Definition:Integration/Integrand",
"Linear Combination of Integrals/Indefinite",
"Definition:Primitive (Calculus)/Constant of Integration",
"Primitive of Reciprocal",
"Primitive of Power",
"Primitive of Power",
"Primiti... |
proofwiki-9163 | Laplace Transform of Second Derivative | Let $f: \R \to \R$ or $\R \to \C$ be a continuous function on any interval of the form $0 \le t \le a$.
Let $f$ be twice differentiable.
Let $f'$ be continuous and $f' '$ piecewise continuous with one-sided limits on said intervals.
Let $f$ and $f'$ be of exponential order.
Let $\laptrans f$ denote the Laplace transfor... | {{Explain|Why does $\laptrans {f' '}$ exist? It needs to be proven}}
{{begin-eqn}}
{{eqn | l = \laptrans {\map {f' '} t}
| r = s \laptrans {\map {f'} t} - \map {f'} 0
| c = Laplace Transform of Derivative
}}
{{eqn | r = s \paren {s \laptrans {\map f t} - \map f 0} - \map {f'} 0
| c = Laplace Transform... | Let $f: \R \to \R$ or $\R \to \C$ be a [[Definition:Continuous Mapping|continuous]] [[Definition:Function|function]] on any interval of the form $0 \le t \le a$.
Let $f$ be [[Definition:Second Derivative|twice differentiable]].
Let $f'$ be [[Definition:Continuous Mapping|continuous]] and $f' '$ [[Definition:Piecewise... | {{Explain|Why does $\laptrans {f' '}$ exist? It needs to be proven}}
{{begin-eqn}}
{{eqn | l = \laptrans {\map {f' '} t}
| r = s \laptrans {\map {f'} t} - \map {f'} 0
| c = [[Laplace Transform of Derivative]]
}}
{{eqn | r = s \paren {s \laptrans {\map f t} - \map f 0} - \map {f'} 0
| c = [[Laplace Tra... | Laplace Transform of Second Derivative | https://proofwiki.org/wiki/Laplace_Transform_of_Second_Derivative | https://proofwiki.org/wiki/Laplace_Transform_of_Second_Derivative | [
"Laplace Transforms of Derivatives"
] | [
"Definition:Continuous Mapping",
"Definition:Function",
"Definition:Derivative/Higher Derivatives/Second Derivative",
"Definition:Continuous Mapping",
"Definition:Piecewise Continuous Function/One-Sided Limits",
"Definition:Exponential Order",
"Definition:Laplace Transform"
] | [
"Laplace Transform of Derivative",
"Laplace Transform of Derivative"
] |
proofwiki-9164 | Euler's Number is Transcendental | Euler's Number $e$ is transcendental. | {{AimForCont}} there exist integers $a_0, \ldots, a_n$ with $a_0 \ne 0$ such that:
:$(1): \quad a_n e^n + a_{n - 1} e^{n - 1} + \cdots + a_0 = 0$
Define $M$, $M_1, \ldots, M_n$ and $\epsilon_1, \ldots, \epsilon_n$ as follows:
{{begin-eqn}}
{{eqn | l = M
| r = \int_0^\infty \frac {x^{p - 1} \sqbrk {\paren {x - 1} ... | [[Definition:Euler's Number|Euler's Number]] $e$ is [[Definition:Transcendental|transcendental]]. | {{AimForCont}} there exist [[Definition:Integer|integers]] $a_0, \ldots, a_n$ with $a_0 \ne 0$ such that:
:$(1): \quad a_n e^n + a_{n - 1} e^{n - 1} + \cdots + a_0 = 0$
Define $M$, $M_1, \ldots, M_n$ and $\epsilon_1, \ldots, \epsilon_n$ as follows:
{{begin-eqn}}
{{eqn | l = M
| r = \int_0^\infty \frac {x^{p -... | Euler's Number is Transcendental/Proof 1 | https://proofwiki.org/wiki/Euler's_Number_is_Transcendental | https://proofwiki.org/wiki/Euler's_Number_is_Transcendental/Proof_1 | [
"Euler's Number is Transcendental",
"Euler's Number",
"Transcendental Number Theory"
] | [
"Definition:Euler's Number",
"Definition:Transcendental"
] | [
"Definition:Integer",
"Definition:Prime Number",
"Definition:Polynomial over Ring",
"Definition:Degree of Polynomial/Field",
"Gamma Function Extends Factorial",
"Prime iff Coprime to all Smaller Positive Integers",
"Euclid's Lemma",
"Common Divisor Divides Difference",
"Definition:Divisor (Algebra)/... |
proofwiki-9165 | Euler's Number is Transcendental | Euler's Number $e$ is transcendental. | {{ProofWanted}}
{{qed}} | [[Definition:Euler's Number|Euler's Number]] $e$ is [[Definition:Transcendental|transcendental]]. | {{ProofWanted}}
{{qed}} | Euler's Number is Transcendental/Proof 2 | https://proofwiki.org/wiki/Euler's_Number_is_Transcendental | https://proofwiki.org/wiki/Euler's_Number_is_Transcendental/Proof_2 | [
"Euler's Number is Transcendental",
"Euler's Number",
"Transcendental Number Theory"
] | [
"Definition:Euler's Number",
"Definition:Transcendental"
] | [] |
proofwiki-9166 | Euler's Number is Transcendental | Euler's Number $e$ is transcendental. | {{ProofWanted}}
{{qed}} | [[Definition:Euler's Number|Euler's Number]] $e$ is [[Definition:Transcendental|transcendental]]. | {{ProofWanted}}
{{qed}} | Euler's Number is Transcendental/Proof 3 | https://proofwiki.org/wiki/Euler's_Number_is_Transcendental | https://proofwiki.org/wiki/Euler's_Number_is_Transcendental/Proof_3 | [
"Euler's Number is Transcendental",
"Euler's Number",
"Transcendental Number Theory"
] | [
"Definition:Euler's Number",
"Definition:Transcendental"
] | [] |
proofwiki-9167 | Laplace Transform of Higher Order Derivatives | {{begin-eqn}}
{{eqn | l = \laptrans {\map {f^{\paren n} } t}
| r = s^n \laptrans {\map f t} - \sum_{j \mathop = 1}^n s^{j - 1} \map {f^{\paren {n - j} } } 0
}}
{{eqn | r = s^n \map F s - s^{n - 1} \, \map f 0 - s^{n - 2} \, \map {f'} 0 - s^{n - 3} \, \map {f' '} 0 - \ldots - s \, \map {f^{\paren {n - 2} } } 0 - \... | The proof proceeds by induction on $n$, the order of the derivative of $f$. | {{begin-eqn}}
{{eqn | l = \laptrans {\map {f^{\paren n} } t}
| r = s^n \laptrans {\map f t} - \sum_{j \mathop = 1}^n s^{j - 1} \map {f^{\paren {n - j} } } 0
}}
{{eqn | r = s^n \map F s - s^{n - 1} \, \map f 0 - s^{n - 2} \, \map {f'} 0 - s^{n - 3} \, \map {f' '} 0 - \ldots - s \, \map {f^{\paren {n - 2} } } 0 - \... | The proof proceeds by [[Principle of Mathematical Induction|induction]] on $n$, the [[Definition:Order of Derivative|order of the derivative]] of $f$. | Laplace Transform of Higher Order Derivatives | https://proofwiki.org/wiki/Laplace_Transform_of_Higher_Order_Derivatives | https://proofwiki.org/wiki/Laplace_Transform_of_Higher_Order_Derivatives | [
"Laplace Transforms of Derivatives"
] | [] | [
"Principle of Mathematical Induction",
"Definition:Derivative/Higher Derivatives/Order of Derivative",
"Principle of Mathematical Induction"
] |
proofwiki-9168 | Laplace Transform of Hyperbolic Cosine | Let $\cosh t$ be the hyperbolic cosine, where $t$ is real.
Let $\laptrans f$ denote the Laplace transform of the real function $f$.
Then:
:$\laptrans {\cosh a t} = \dfrac s {s^2 - a^2}$
where $a \in \R_{>0}$ is constant, and $\map \Re s > a$. | {{begin-eqn}}
{{eqn | l = \laptrans {\cosh at}
| r = \laptrans {\frac {e^{at} + e^{-at} } 2}
| c = {{Defof|Hyperbolic Cosine}}
}}
{{eqn | r = \frac 1 2 \paren {\laptrans {e^{at} } + \laptrans {e^{-at} } }
| c = Linear Combination of Laplace Transforms
}}
{{eqn | r = \frac 1 2 \paren {\frac 1 {s - a} +... | Let $\cosh t$ be the [[Definition:Hyperbolic Cosine|hyperbolic cosine]], where $t$ is [[Definition:Real Number|real]].
Let $\laptrans f$ denote the [[Definition:Laplace Transform|Laplace transform]] of the [[Definition:Real Function|real function]] $f$.
Then:
:$\laptrans {\cosh a t} = \dfrac s {s^2 - a^2}$
where $... | {{begin-eqn}}
{{eqn | l = \laptrans {\cosh at}
| r = \laptrans {\frac {e^{at} + e^{-at} } 2}
| c = {{Defof|Hyperbolic Cosine}}
}}
{{eqn | r = \frac 1 2 \paren {\laptrans {e^{at} } + \laptrans {e^{-at} } }
| c = [[Linear Combination of Laplace Transforms]]
}}
{{eqn | r = \frac 1 2 \paren {\frac 1 {s - ... | Laplace Transform of Hyperbolic Cosine/Proof 2 | https://proofwiki.org/wiki/Laplace_Transform_of_Hyperbolic_Cosine | https://proofwiki.org/wiki/Laplace_Transform_of_Hyperbolic_Cosine/Proof_2 | [
"Laplace Transform of Hyperbolic Cosine",
"Laplace Transforms involving Hyperbolic Cosine Function",
"Laplace Transforms involving Hyperbolic Functions",
"Examples of Laplace Transforms",
"Hyperbolic Cosine Function"
] | [
"Definition:Hyperbolic Cosine",
"Definition:Real Number",
"Definition:Laplace Transform",
"Definition:Real Function",
"Definition:Constant"
] | [
"Linear Combination of Laplace Transforms",
"Laplace Transform of Exponential"
] |
proofwiki-9169 | Laplace Transform of Hyperbolic Cosine | Let $\cosh t$ be the hyperbolic cosine, where $t$ is real.
Let $\laptrans f$ denote the Laplace transform of the real function $f$.
Then:
:$\laptrans {\cosh a t} = \dfrac s {s^2 - a^2}$
where $a \in \R_{>0}$ is constant, and $\map \Re s > a$. | {{begin-eqn}}
{{eqn | l = \laptrans {\cosh a t}
| r = \laptrans {\frac {e^{a t} + e^{-a t} } 2}
| c = {{Defof|Hyperbolic Cosine}}
}}
{{eqn | r = \int_0^{\to +\infty} e^{-s t} \paren {\frac {e^{a t} + e^{-a t} } 2} \rd t
| c = {{Defof|Laplace Transform}}
}}
{{eqn | r = \dfrac 1 2 \int_0^{\to +\infty} e... | Let $\cosh t$ be the [[Definition:Hyperbolic Cosine|hyperbolic cosine]], where $t$ is [[Definition:Real Number|real]].
Let $\laptrans f$ denote the [[Definition:Laplace Transform|Laplace transform]] of the [[Definition:Real Function|real function]] $f$.
Then:
:$\laptrans {\cosh a t} = \dfrac s {s^2 - a^2}$
where $... | {{begin-eqn}}
{{eqn | l = \laptrans {\cosh a t}
| r = \laptrans {\frac {e^{a t} + e^{-a t} } 2}
| c = {{Defof|Hyperbolic Cosine}}
}}
{{eqn | r = \int_0^{\to +\infty} e^{-s t} \paren {\frac {e^{a t} + e^{-a t} } 2} \rd t
| c = {{Defof|Laplace Transform}}
}}
{{eqn | r = \dfrac 1 2 \int_0^{\to +\infty} e... | Laplace Transform of Hyperbolic Cosine/Proof 3 | https://proofwiki.org/wiki/Laplace_Transform_of_Hyperbolic_Cosine | https://proofwiki.org/wiki/Laplace_Transform_of_Hyperbolic_Cosine/Proof_3 | [
"Laplace Transform of Hyperbolic Cosine",
"Laplace Transforms involving Hyperbolic Cosine Function",
"Laplace Transforms involving Hyperbolic Functions",
"Examples of Laplace Transforms",
"Hyperbolic Cosine Function"
] | [
"Definition:Hyperbolic Cosine",
"Definition:Real Number",
"Definition:Laplace Transform",
"Definition:Real Function",
"Definition:Constant"
] | [
"Linear Combination of Laplace Transforms",
"Laplace Transform of Exponential"
] |
proofwiki-9170 | Laplace Transform of Hyperbolic Sine | Let $\sinh t$ be the hyperbolic sine, where $t$ is real.
Let $\laptrans f$ denote the Laplace transform of the real function $f$.
Then:
:$\laptrans {\sinh a t} = \dfrac a {s^2 - a^2}$
where $a \in \R_{>0}$ is constant, and $\map \Re s > a$. | {{begin-eqn}}
{{eqn | l = \laptrans {\sinh {a t} }
| r = \int_0^{\to +\infty} e^{-s t} \sinh {a t} \rd t
| c = {{Defof|Laplace Transform}}
}}
{{eqn | r = \intlimits {\frac {e^{-s t} \paren {-s \sinh a t - a \cosh a t} } {\paren {-s}^2 - a^2} } {t \mathop = 0} {t \mathop \to +\infty}
| c = Primitive of... | Let $\sinh t$ be the [[Definition:Hyperbolic Sine|hyperbolic sine]], where $t$ is [[Definition:Real Number|real]].
Let $\laptrans f$ denote the [[Definition:Laplace Transform|Laplace transform]] of the [[Definition:Real Function|real function]] $f$.
Then:
:$\laptrans {\sinh a t} = \dfrac a {s^2 - a^2}$
where $a \i... | {{begin-eqn}}
{{eqn | l = \laptrans {\sinh {a t} }
| r = \int_0^{\to +\infty} e^{-s t} \sinh {a t} \rd t
| c = {{Defof|Laplace Transform}}
}}
{{eqn | r = \intlimits {\frac {e^{-s t} \paren {-s \sinh a t - a \cosh a t} } {\paren {-s}^2 - a^2} } {t \mathop = 0} {t \mathop \to +\infty}
| c = [[Primitive ... | Laplace Transform of Hyperbolic Sine/Proof 1 | https://proofwiki.org/wiki/Laplace_Transform_of_Hyperbolic_Sine | https://proofwiki.org/wiki/Laplace_Transform_of_Hyperbolic_Sine/Proof_1 | [
"Laplace Transform of Hyperbolic Sine",
"Laplace Transforms involving Hyperbolic Sine Function",
"Laplace Transforms involving Hyperbolic Functions",
"Examples of Laplace Transforms",
"Hyperbolic Sine Function"
] | [
"Definition:Hyperbolic Sine",
"Definition:Real Number",
"Definition:Laplace Transform",
"Definition:Real Function",
"Definition:Constant"
] | [
"Primitive of Exponential of a x by Hyperbolic Sine of b x",
"Exponential Tends to Zero and Infinity",
"Exponential of Zero",
"Hyperbolic Sine of Zero is Zero",
"Hyperbolic Cosine of Zero is One"
] |
proofwiki-9171 | Laplace Transform of Hyperbolic Sine | Let $\sinh t$ be the hyperbolic sine, where $t$ is real.
Let $\laptrans f$ denote the Laplace transform of the real function $f$.
Then:
:$\laptrans {\sinh a t} = \dfrac a {s^2 - a^2}$
where $a \in \R_{>0}$ is constant, and $\map \Re s > a$. | {{begin-eqn}}
{{eqn | l = \laptrans {\sinh a t}
| r = \laptrans {\frac {e^{a t} - e^{-a t} } 2}
| c = {{Defof|Hyperbolic Sine}}
}}
{{eqn | r = \frac 1 2 \paren {\laptrans {e^{at} } - \laptrans {e^{-a t} } }
| c = Linear Combination of Laplace Transforms
}}
{{eqn | r = \frac 1 2 \paren {\frac 1 {s-a} -... | Let $\sinh t$ be the [[Definition:Hyperbolic Sine|hyperbolic sine]], where $t$ is [[Definition:Real Number|real]].
Let $\laptrans f$ denote the [[Definition:Laplace Transform|Laplace transform]] of the [[Definition:Real Function|real function]] $f$.
Then:
:$\laptrans {\sinh a t} = \dfrac a {s^2 - a^2}$
where $a \i... | {{begin-eqn}}
{{eqn | l = \laptrans {\sinh a t}
| r = \laptrans {\frac {e^{a t} - e^{-a t} } 2}
| c = {{Defof|Hyperbolic Sine}}
}}
{{eqn | r = \frac 1 2 \paren {\laptrans {e^{at} } - \laptrans {e^{-a t} } }
| c = [[Linear Combination of Laplace Transforms]]
}}
{{eqn | r = \frac 1 2 \paren {\frac 1 {s-... | Laplace Transform of Hyperbolic Sine/Proof 2 | https://proofwiki.org/wiki/Laplace_Transform_of_Hyperbolic_Sine | https://proofwiki.org/wiki/Laplace_Transform_of_Hyperbolic_Sine/Proof_2 | [
"Laplace Transform of Hyperbolic Sine",
"Laplace Transforms involving Hyperbolic Sine Function",
"Laplace Transforms involving Hyperbolic Functions",
"Examples of Laplace Transforms",
"Hyperbolic Sine Function"
] | [
"Definition:Hyperbolic Sine",
"Definition:Real Number",
"Definition:Laplace Transform",
"Definition:Real Function",
"Definition:Constant"
] | [
"Linear Combination of Laplace Transforms",
"Laplace Transform of Exponential"
] |
proofwiki-9172 | Laplace Transform of Hyperbolic Sine | Let $\sinh t$ be the hyperbolic sine, where $t$ is real.
Let $\laptrans f$ denote the Laplace transform of the real function $f$.
Then:
:$\laptrans {\sinh a t} = \dfrac a {s^2 - a^2}$
where $a \in \R_{>0}$ is constant, and $\map \Re s > a$. | {{begin-eqn}}
{{eqn | l = \laptrans {\sinh a t}
| r = \laptrans {\frac {e^{a t} - e^{-a t} } 2}
| c = {{Defof|Hyperbolic Sine}}
}}
{{eqn | r = \int_0^{\to +\infty} e^{-s t} \paren {\frac {e^{a t} - e^{-a t} } 2} \rd t
| c = {{Defof|Laplace Transform}}
}}
{{eqn | r = \dfrac 1 2 \int_0^{\to +\infty} e^{... | Let $\sinh t$ be the [[Definition:Hyperbolic Sine|hyperbolic sine]], where $t$ is [[Definition:Real Number|real]].
Let $\laptrans f$ denote the [[Definition:Laplace Transform|Laplace transform]] of the [[Definition:Real Function|real function]] $f$.
Then:
:$\laptrans {\sinh a t} = \dfrac a {s^2 - a^2}$
where $a \i... | {{begin-eqn}}
{{eqn | l = \laptrans {\sinh a t}
| r = \laptrans {\frac {e^{a t} - e^{-a t} } 2}
| c = {{Defof|Hyperbolic Sine}}
}}
{{eqn | r = \int_0^{\to +\infty} e^{-s t} \paren {\frac {e^{a t} - e^{-a t} } 2} \rd t
| c = {{Defof|Laplace Transform}}
}}
{{eqn | r = \dfrac 1 2 \int_0^{\to +\infty} e^{... | Laplace Transform of Hyperbolic Sine/Proof 3 | https://proofwiki.org/wiki/Laplace_Transform_of_Hyperbolic_Sine | https://proofwiki.org/wiki/Laplace_Transform_of_Hyperbolic_Sine/Proof_3 | [
"Laplace Transform of Hyperbolic Sine",
"Laplace Transforms involving Hyperbolic Sine Function",
"Laplace Transforms involving Hyperbolic Functions",
"Examples of Laplace Transforms",
"Hyperbolic Sine Function"
] | [
"Definition:Hyperbolic Sine",
"Definition:Real Number",
"Definition:Laplace Transform",
"Definition:Real Function",
"Definition:Constant"
] | [
"Linear Combination of Laplace Transforms",
"Laplace Transform of Exponential"
] |
proofwiki-9173 | Primitive of Power of a x + b | :$\ds \int \paren {a x + b}^n \rd x = \frac {\paren {a x + b}^{n + 1} } {\paren {n + 1} a} + C$ | Let $u = a x + b$.
Then:
{{begin-eqn}}
{{eqn | l = \int \paren {a x + b}^n \rd x
| r = \frac 1 a \int u^n \rd u
| c = Primitive of Function of $a x + b$
}}
{{eqn | r = \frac 1 a \frac {u^{n + 1} } {n + 1} + C
| c = Primitive of Power
}}
{{eqn | r = \frac {\paren {a x + b}^{n + 1} } {\paren {n + 1} a} ... | :$\ds \int \paren {a x + b}^n \rd x = \frac {\paren {a x + b}^{n + 1} } {\paren {n + 1} a} + C$ | Let $u = a x + b$.
Then:
{{begin-eqn}}
{{eqn | l = \int \paren {a x + b}^n \rd x
| r = \frac 1 a \int u^n \rd u
| c = [[Primitive of Function of a x + b|Primitive of Function of $a x + b$]]
}}
{{eqn | r = \frac 1 a \frac {u^{n + 1} } {n + 1} + C
| c = [[Primitive of Power]]
}}
{{eqn | r = \frac {\pa... | Primitive of Power of a x + b/Proof 1 | https://proofwiki.org/wiki/Primitive_of_Power_of_a_x_+_b | https://proofwiki.org/wiki/Primitive_of_Power_of_a_x_+_b/Proof_1 | [
"Primitive of Power of a x + b",
"Primitives involving a x + b"
] | [] | [
"Primitive of Function of a x + b",
"Primitive of Power"
] |
proofwiki-9174 | Primitive of Power of a x + b | :$\ds \int \paren {a x + b}^n \rd x = \frac {\paren {a x + b}^{n + 1} } {\paren {n + 1} a} + C$ | Let $u = a x + b$.
Then:
:$\dfrac {\d u} {\d x} = a$
Then:
{{begin-eqn}}
{{eqn | l = \int \paren {a x + b}^n \rd x
| r = \int \dfrac {u^n} a \rd u
| c = Integration by Substitution
}}
{{eqn | r = \dfrac 1 a \dfrac {u^{n + 1} } {n + 1}
| c = Primitive of Power
}}
{{eqn | r = \frac {\paren {a x + b}^{n ... | :$\ds \int \paren {a x + b}^n \rd x = \frac {\paren {a x + b}^{n + 1} } {\paren {n + 1} a} + C$ | Let $u = a x + b$.
Then:
:$\dfrac {\d u} {\d x} = a$
Then:
{{begin-eqn}}
{{eqn | l = \int \paren {a x + b}^n \rd x
| r = \int \dfrac {u^n} a \rd u
| c = [[Integration by Substitution]]
}}
{{eqn | r = \dfrac 1 a \dfrac {u^{n + 1} } {n + 1}
| c = [[Primitive of Power]]
}}
{{eqn | r = \frac {\paren {a ... | Primitive of Power of a x + b/Proof 2 | https://proofwiki.org/wiki/Primitive_of_Power_of_a_x_+_b | https://proofwiki.org/wiki/Primitive_of_Power_of_a_x_+_b/Proof_2 | [
"Primitive of Power of a x + b",
"Primitives involving a x + b"
] | [] | [
"Integration by Substitution",
"Primitive of Power"
] |
proofwiki-9175 | Primitive of Power of a x + b | :$\ds \int \paren {a x + b}^n \rd x = \frac {\paren {a x + b}^{n + 1} } {\paren {n + 1} a} + C$ | {{begin-eqn}}
{{eqn | l = \map {\dfrac \d {\d x} } {\frac {\paren {a x + b}^{n + 1} } {\paren {n + 1} a} }
| r = \dfrac {\paren {n + 1} \paren {a x + b}^n} {\paren {n + 1} a} \map {\dfrac \d {\d x} } {a x + b}
| c = Power Rule for Derivatives, Chain Rule for Derivatives
}}
{{eqn | r = \dfrac {a \paren {n + ... | :$\ds \int \paren {a x + b}^n \rd x = \frac {\paren {a x + b}^{n + 1} } {\paren {n + 1} a} + C$ | {{begin-eqn}}
{{eqn | l = \map {\dfrac \d {\d x} } {\frac {\paren {a x + b}^{n + 1} } {\paren {n + 1} a} }
| r = \dfrac {\paren {n + 1} \paren {a x + b}^n} {\paren {n + 1} a} \map {\dfrac \d {\d x} } {a x + b}
| c = [[Power Rule for Derivatives]], [[Chain Rule for Derivatives]]
}}
{{eqn | r = \dfrac {a \par... | Primitive of Power of a x + b/Proof 3 | https://proofwiki.org/wiki/Primitive_of_Power_of_a_x_+_b | https://proofwiki.org/wiki/Primitive_of_Power_of_a_x_+_b/Proof_3 | [
"Primitive of Power of a x + b",
"Primitives involving a x + b"
] | [] | [
"Power Rule for Derivatives",
"Derivative of Composite Function",
"Power Rule for Derivatives",
"Definition:Primitive (Calculus)"
] |
proofwiki-9176 | Primitive of x by Power of a x + b | :$\ds \int x \paren {a x + b}^n \rd x = \frac {\paren {a x + b}^{n + 2} } {\paren {n + 2} a^2} - \frac {b \paren {a x + b}^{n + 1} } {\paren {n + 1} a^2} + C$ | Let $u = a x + b$.
Then:
{{begin-eqn}}
{{eqn | l = x
| r = \frac {u - b} a
| c =
}}
{{eqn | l = \frac {\d u} {\d x}
| r = \frac 1 a
| c =
}}
{{end-eqn}}
Then:
{{begin-eqn}}
{{eqn | l = \int x \paren {a x + b}^n \rd x
| r = \frac 1 a \int \frac {u - b} a u^n \rd u
| c = Integration ... | :$\ds \int x \paren {a x + b}^n \rd x = \frac {\paren {a x + b}^{n + 2} } {\paren {n + 2} a^2} - \frac {b \paren {a x + b}^{n + 1} } {\paren {n + 1} a^2} + C$ | Let $u = a x + b$.
Then:
{{begin-eqn}}
{{eqn | l = x
| r = \frac {u - b} a
| c =
}}
{{eqn | l = \frac {\d u} {\d x}
| r = \frac 1 a
| c =
}}
{{end-eqn}}
Then:
{{begin-eqn}}
{{eqn | l = \int x \paren {a x + b}^n \rd x
| r = \frac 1 a \int \frac {u - b} a u^n \rd u
| c = [[Integr... | Primitive of x by Power of a x + b | https://proofwiki.org/wiki/Primitive_of_x_by_Power_of_a_x_+_b | https://proofwiki.org/wiki/Primitive_of_x_by_Power_of_a_x_+_b | [
"Primitive of x by Power of a x + b",
"Primitives involving a x + b"
] | [] | [
"Integration by Substitution",
"Linear Combination of Integrals/Indefinite",
"Primitive of Power"
] |
proofwiki-9177 | Primitive of x squared by Power of a x + b | :$\ds \int x^2 \paren {a x + b}^n \rd x = \frac {\paren {a x + b}^{n + 3} } {\paren {n + 3} a^3} - \frac {2 b \paren {a x + b}^{n + 2} } {\paren {n + 2} a^3} + \frac {b^2 \paren {a x + b}^{n + 1} } {\paren {n + 1} a^3} + C$ | Let $u = a x + b$.
Then:
{{begin-eqn}}
{{eqn | l = x
| r = \frac {u - b} a
| c =
}}
{{eqn | l = \frac {\d u} {\d x}
| r = \frac 1 a
| c =
}}
{{end-eqn}}
Then:
{{begin-eqn}}
{{eqn | l = \int x \paren {a x + b}^n \rd x
| r = \int \frac 1 a \paren {\frac {u - b} a}^2 u^n \rd u
| c = I... | :$\ds \int x^2 \paren {a x + b}^n \rd x = \frac {\paren {a x + b}^{n + 3} } {\paren {n + 3} a^3} - \frac {2 b \paren {a x + b}^{n + 2} } {\paren {n + 2} a^3} + \frac {b^2 \paren {a x + b}^{n + 1} } {\paren {n + 1} a^3} + C$ | Let $u = a x + b$.
Then:
{{begin-eqn}}
{{eqn | l = x
| r = \frac {u - b} a
| c =
}}
{{eqn | l = \frac {\d u} {\d x}
| r = \frac 1 a
| c =
}}
{{end-eqn}}
Then:
{{begin-eqn}}
{{eqn | l = \int x \paren {a x + b}^n \rd x
| r = \int \frac 1 a \paren {\frac {u - b} a}^2 u^n \rd u
| c... | Primitive of x squared by Power of a x + b | https://proofwiki.org/wiki/Primitive_of_x_squared_by_Power_of_a_x_+_b | https://proofwiki.org/wiki/Primitive_of_x_squared_by_Power_of_a_x_+_b | [
"Primitives involving a x + b"
] | [] | [
"Integration by Substitution",
"Square of Difference",
"Linear Combination of Integrals/Indefinite",
"Primitive of Power"
] |
proofwiki-9178 | Primitive of Power of x by Power of a x + b/Decrement of Power of a x + b | :$\ds \int x^m \paren {a x + b}^n \rd x = \frac {x^{m + 1} \paren {a x + b}^n} {m + n + 1} + \frac {n b} {m + n + 1} \int x^m \paren {a x + b}^{n - 1} \rd x$ | Let $s \in \Z$.
{{begin-eqn}}
{{eqn | l = v
| r = x^s
| c =
}}
{{eqn | n = 1
| ll= \leadsto
| l = \frac {\d v} {\d x}
| r = s x^{s - 1}
| c = Power Rule for Derivatives
}}
{{end-eqn}}
Let $u \dfrac {\d v} {\d x} = x^m \paren {a x + b}^n$.
Then:
{{begin-eqn}}
{{eqn | l = u
| r... | :$\ds \int x^m \paren {a x + b}^n \rd x = \frac {x^{m + 1} \paren {a x + b}^n} {m + n + 1} + \frac {n b} {m + n + 1} \int x^m \paren {a x + b}^{n - 1} \rd x$ | Let $s \in \Z$.
{{begin-eqn}}
{{eqn | l = v
| r = x^s
| c =
}}
{{eqn | n = 1
| ll= \leadsto
| l = \frac {\d v} {\d x}
| r = s x^{s - 1}
| c = [[Power Rule for Derivatives]]
}}
{{end-eqn}}
Let $u \dfrac {\d v} {\d x} = x^m \paren {a x + b}^n$.
Then:
{{begin-eqn}}
{{eqn | l = u
... | Primitive of Power of x by Power of a x + b/Decrement of Power of a x + b/Proof 1 | https://proofwiki.org/wiki/Primitive_of_Power_of_x_by_Power_of_a_x_+_b/Decrement_of_Power_of_a_x_+_b | https://proofwiki.org/wiki/Primitive_of_Power_of_x_by_Power_of_a_x_+_b/Decrement_of_Power_of_a_x_+_b/Proof_1 | [
"Primitive of Power of x by Power of a x + b"
] | [] | [
"Power Rule for Derivatives",
"Primitive of Power of a x + b",
"Product Rule for Derivatives",
"Integration by Parts"
] |
proofwiki-9179 | Primitive of Power of x by Power of a x + b/Decrement of Power of a x + b | :$\ds \int x^m \paren {a x + b}^n \rd x = \frac {x^{m + 1} \paren {a x + b}^n} {m + n + 1} + \frac {n b} {m + n + 1} \int x^m \paren {a x + b}^{n - 1} \rd x$ | From Primitive of Power of $a x + b$ by Power of $p x + q$: Decrement of Power:
:$\ds \int \paren {a x + b}^m \paren {p x + q}^n \rd x = \frac {\paren {a x + b}^{m + 1} \paren {p x + q}^n} {\paren {m + n + 1} a} - \frac {n \paren {b p - a q} } {\paren {m + n + 1} a} \int \paren {a x + b}^m \paren {p x + q}^{n - 1} \rd ... | :$\ds \int x^m \paren {a x + b}^n \rd x = \frac {x^{m + 1} \paren {a x + b}^n} {m + n + 1} + \frac {n b} {m + n + 1} \int x^m \paren {a x + b}^{n - 1} \rd x$ | From [[Primitive of Power of a x + b by Power of p x + q/Decrement of Power|Primitive of Power of $a x + b$ by Power of $p x + q$: Decrement of Power]]:
:$\ds \int \paren {a x + b}^m \paren {p x + q}^n \rd x = \frac {\paren {a x + b}^{m + 1} \paren {p x + q}^n} {\paren {m + n + 1} a} - \frac {n \paren {b p - a q} } {\p... | Primitive of Power of x by Power of a x + b/Decrement of Power of a x + b/Proof 2 | https://proofwiki.org/wiki/Primitive_of_Power_of_x_by_Power_of_a_x_+_b/Decrement_of_Power_of_a_x_+_b | https://proofwiki.org/wiki/Primitive_of_Power_of_x_by_Power_of_a_x_+_b/Decrement_of_Power_of_a_x_+_b/Proof_2 | [
"Primitive of Power of x by Power of a x + b"
] | [] | [
"Primitive of Power of a x + b by Power of p x + q/Decrement of Power"
] |
proofwiki-9180 | Distance from Subset to Element | Let $\struct {M, d}$ be a metric space.
Let $S \subseteq M$ be a subset of $M$.
Let $s \in S$.
Then:
:$\map d {s, S} = 0$
where $\map d {s, S}$ denotes the distance between $s$ and $S$. | By Distance between Element and Subset is Nonnegative:
:$\map d {s, S} \ge 0$
Also, because $s \in S$, it follows that:
:$\map d {s, S} \le \map d {s, s} = 0$
Hence the result.
{{qed}}
Category:Distance Function
g2dc24h1w1u2grawyqcpwoyoz5ziw3y | Let $\struct {M, d}$ be a [[Definition:Metric Space|metric space]].
Let $S \subseteq M$ be a [[Definition:Subset|subset]] of $M$.
Let $s \in S$.
Then:
:$\map d {s, S} = 0$
where $\map d {s, S}$ denotes the [[Definition:Distance between Element and Subset of Metric Space|distance between $s$ and $S$]]. | By [[Distance between Element and Subset is Nonnegative]]:
:$\map d {s, S} \ge 0$
Also, because $s \in S$, it follows that:
:$\map d {s, S} \le \map d {s, s} = 0$
Hence the result.
{{qed}}
[[Category:Distance Function]]
g2dc24h1w1u2grawyqcpwoyoz5ziw3y | Distance from Subset to Element | https://proofwiki.org/wiki/Distance_from_Subset_to_Element | https://proofwiki.org/wiki/Distance_from_Subset_to_Element | [
"Distance Function"
] | [
"Definition:Metric Space",
"Definition:Subset",
"Definition:Distance/Sets/Metric Spaces"
] | [
"Distance between Element and Subset is Nonnegative",
"Category:Distance Function"
] |
proofwiki-9181 | Primitive of Power of x by Power of a x + b/Decrement of Power of x | :$\ds \int x^m \paren {a x + b}^n \rd x = \frac {x^m \paren {a x + b}^{n + 1} } {\paren {m + n + 1} a} - \frac {m b} {\paren {m + n + 1} a} \int x^{m - 1} \paren {a x + b}^n \rd x$ | Let $s \in \Z$.
{{begin-eqn}}
{{eqn | l = v
| r = \paren {a x + b}^s
| c =
}}
{{eqn | n = 1
| ll= \leadsto
| l = \frac {\d v} {\d x}
| r = a s \paren {a x + b}^{s - 1}
| c = Power Rule for Derivatives and Derivatives of Function of $a x + b$
}}
{{end-eqn}}
Let $u \dfrac {\d v} {\d ... | :$\ds \int x^m \paren {a x + b}^n \rd x = \frac {x^m \paren {a x + b}^{n + 1} } {\paren {m + n + 1} a} - \frac {m b} {\paren {m + n + 1} a} \int x^{m - 1} \paren {a x + b}^n \rd x$ | Let $s \in \Z$.
{{begin-eqn}}
{{eqn | l = v
| r = \paren {a x + b}^s
| c =
}}
{{eqn | n = 1
| ll= \leadsto
| l = \frac {\d v} {\d x}
| r = a s \paren {a x + b}^{s - 1}
| c = [[Power Rule for Derivatives]] and [[Derivatives of Function of a x + b|Derivatives of Function of $a x + b... | Primitive of Power of x by Power of a x + b/Decrement of Power of x/Proof 1 | https://proofwiki.org/wiki/Primitive_of_Power_of_x_by_Power_of_a_x_+_b/Decrement_of_Power_of_x | https://proofwiki.org/wiki/Primitive_of_Power_of_x_by_Power_of_a_x_+_b/Decrement_of_Power_of_x/Proof_1 | [
"Primitive of Power of x by Power of a x + b"
] | [] | [
"Power Rule for Derivatives",
"Derivatives of Function of a x + b",
"Product Rule for Derivatives",
"Integration by Parts"
] |
proofwiki-9182 | Primitive of Power of x by Power of a x + b/Decrement of Power of x | :$\ds \int x^m \paren {a x + b}^n \rd x = \frac {x^m \paren {a x + b}^{n + 1} } {\paren {m + n + 1} a} - \frac {m b} {\paren {m + n + 1} a} \int x^{m - 1} \paren {a x + b}^n \rd x$ | From Primitive of Power of $a x + b$ by Power of $p x + q$: Decrement of Power:
:$\ds \int \paren {a x + b}^m \paren {p x + q}^n \rd x = \frac {\paren {a x + b}^{m + 1} \paren {p x + q}^n} {\paren {m + n + 1} a} - \frac {n \paren {b p - a q} } {\paren {m + n + 1} a} \int \paren {a x + b}^m \paren {p x + q}^{n - 1} \rd ... | :$\ds \int x^m \paren {a x + b}^n \rd x = \frac {x^m \paren {a x + b}^{n + 1} } {\paren {m + n + 1} a} - \frac {m b} {\paren {m + n + 1} a} \int x^{m - 1} \paren {a x + b}^n \rd x$ | From [[Primitive of Power of a x + b by Power of p x + q/Decrement of Power|Primitive of Power of $a x + b$ by Power of $p x + q$: Decrement of Power]]:
:$\ds \int \paren {a x + b}^m \paren {p x + q}^n \rd x = \frac {\paren {a x + b}^{m + 1} \paren {p x + q}^n} {\paren {m + n + 1} a} - \frac {n \paren {b p - a q} } {\p... | Primitive of Power of x by Power of a x + b/Decrement of Power of x/Proof 2 | https://proofwiki.org/wiki/Primitive_of_Power_of_x_by_Power_of_a_x_+_b/Decrement_of_Power_of_x | https://proofwiki.org/wiki/Primitive_of_Power_of_x_by_Power_of_a_x_+_b/Decrement_of_Power_of_x/Proof_2 | [
"Primitive of Power of x by Power of a x + b"
] | [] | [
"Primitive of Power of a x + b by Power of p x + q/Decrement of Power"
] |
proofwiki-9183 | Distance from Subset to Supremum | Let $S \subseteq \R$ be a subset of the real numbers.
Suppose that the supremum $\sup S$ of $S$ exists.
Then:
:$\map d {\sup S, S} = 0$
where $\map d {\sup S, S}$ is the distance between $\sup S$ and $S$. | By Distance between Element and Subset is Nonnegative:
:$\map d {\sup S, S} \ge 0$
By definition of supremum:
:$\forall \epsilon > 0: \exists s \in S: \map d {\sup S, s} < \epsilon$
meaning that, by nature of the infimum and the definition of $\map d {\sup S, S}$:
:$\forall \epsilon > 0: \map d {\sup S, S} < \epsilon$
... | Let $S \subseteq \R$ be a [[Definition:Subset|subset]] of the [[Definition:Real Numbers|real numbers]].
Suppose that the [[Definition:Supremum of Subset of Real Numbers|supremum]] $\sup S$ of $S$ exists.
Then:
:$\map d {\sup S, S} = 0$
where $\map d {\sup S, S}$ is the [[Definition:Distance between Element and Sub... | By [[Distance between Element and Subset is Nonnegative]]:
:$\map d {\sup S, S} \ge 0$
By definition of [[Definition:Supremum of Subset of Real Numbers|supremum]]:
:$\forall \epsilon > 0: \exists s \in S: \map d {\sup S, s} < \epsilon$
meaning that, by nature of the [[Definition:Infimum of Subset of Real Numbers|in... | Distance from Subset to Supremum | https://proofwiki.org/wiki/Distance_from_Subset_to_Supremum | https://proofwiki.org/wiki/Distance_from_Subset_to_Supremum | [
"Real Analysis",
"Distance Function"
] | [
"Definition:Subset",
"Definition:Real Number",
"Definition:Supremum of Set/Real Numbers",
"Definition:Distance/Sets/Real Numbers"
] | [
"Distance between Element and Subset is Nonnegative",
"Definition:Supremum of Set/Real Numbers",
"Definition:Infimum of Set/Real Numbers",
"Definition:Distance/Sets/Real Numbers"
] |
proofwiki-9184 | Distance between Element and Subset is Nonnegative | Let $\struct {M, d}$ be a metric space.
Let $x \in M$ and $S \subseteq M$.
Then:
:$\map d {x, S} \ge 0$
where $\map d {x, S}$ is the distance between $x$ and $S$. | By definition of the distance between $x$ and $S$:
:$\map d {x, S} = \ds \inf_{s \mathop \in S} \map d {x, s}$
From the metric space axioms:
:$\forall s \in M: \map d {x, s} \ge 0$
Hence by the nature of the infimum:
:$\map d {x, S} \ge 0$
as desired.
{{qed}}
Category:Distance Function
70b2wntin5ezf0alcdgs8v9lmbkscm4 | Let $\struct {M, d}$ be a [[Definition:Metric Space|metric space]].
Let $x \in M$ and $S \subseteq M$.
Then:
:$\map d {x, S} \ge 0$
where $\map d {x, S}$ is the [[Definition:Distance between Element and Subset of Metric Space|distance between $x$ and $S$]]. | By definition of the [[Definition:Distance between Element and Subset of Metric Space|distance between $x$ and $S$]]:
:$\map d {x, S} = \ds \inf_{s \mathop \in S} \map d {x, s}$
From the [[Axiom:Metric Space Axioms|metric space axioms]]:
:$\forall s \in M: \map d {x, s} \ge 0$
Hence by the nature of the [[Definitio... | Distance between Element and Subset is Nonnegative | https://proofwiki.org/wiki/Distance_between_Element_and_Subset_is_Nonnegative | https://proofwiki.org/wiki/Distance_between_Element_and_Subset_is_Nonnegative | [
"Distance Function"
] | [
"Definition:Metric Space",
"Definition:Distance/Sets/Metric Spaces"
] | [
"Definition:Distance/Sets/Metric Spaces",
"Axiom:Metric Space Axioms",
"Definition:Infimum of Set/Real Numbers",
"Category:Distance Function"
] |
proofwiki-9185 | Distance from Subset to Infimum | Let $S \subseteq \R$ be a subset of the real numbers.
Suppose that the infimum $\inf S$ of $S$ exists.
Then:
:$\map d {\inf S, S} = 0$
where $\map d {\inf S, S}$ is the distance between $\inf S$ and $S$. | By Distance between Element and Subset is Nonnegative:
:$\map d {\inf S, S} \ge 0$
By definition of infimum:
:$\forall \epsilon > 0: \exists s \in S: \map d {\inf S, s} < \epsilon$
meaning that, by nature of the infimum and the definition of $\map d {\inf S, S}$:
:$\forall \epsilon > 0: \map d {\inf S, S} < \epsilon$
T... | Let $S \subseteq \R$ be a [[Definition:Subset|subset]] of the [[Definition:Real Numbers|real numbers]].
Suppose that the [[Definition:Infimum of Subset of Real Numbers|infimum]] $\inf S$ of $S$ exists.
Then:
:$\map d {\inf S, S} = 0$
where $\map d {\inf S, S}$ is the [[Definition:Distance between Element and Subse... | By [[Distance between Element and Subset is Nonnegative]]:
:$\map d {\inf S, S} \ge 0$
By definition of [[Definition:Infimum of Subset of Real Numbers|infimum]]:
:$\forall \epsilon > 0: \exists s \in S: \map d {\inf S, s} < \epsilon$
meaning that, by nature of the [[Definition:Infimum of Subset of Real Numbers|infi... | Distance from Subset to Infimum | https://proofwiki.org/wiki/Distance_from_Subset_to_Infimum | https://proofwiki.org/wiki/Distance_from_Subset_to_Infimum | [
"Real Analysis"
] | [
"Definition:Subset",
"Definition:Real Number",
"Definition:Infimum of Set/Real Numbers",
"Definition:Distance/Sets/Real Numbers"
] | [
"Distance between Element and Subset is Nonnegative",
"Definition:Infimum of Set/Real Numbers",
"Definition:Infimum of Set/Real Numbers",
"Definition:Distance/Sets/Real Numbers",
"Category:Real Analysis"
] |
proofwiki-9186 | Primitive of Power of x by Power of a x + b/Increment of Power of a x + b | :$\ds \int x^m \paren {a x + b}^n \rd x = \frac {-x^{m + 1} \paren {a x + b}^{n + 1} } {\paren {n + 1} b} + \frac {m + n + 2} {\paren {n + 1} b} \int x^m \paren {a x + b}^{n + 1} \rd x$ | From Primitive of Power of $x$ by Power of $a x + b$: Decrement of Power of $x$:
:$\ds \int x^m \paren {a x + b}^n \rd x = \frac {x^{m + 1} \paren {a x + b}^n} {m + n + 1} + \frac {n b} {m + n + 1} \int x^m \paren {a x + b}^{n - 1} \rd x$
Substituting $n + 1$ for $n$:
{{begin-eqn}}
{{eqn | l = \int x^m \paren {a x + b}... | :$\ds \int x^m \paren {a x + b}^n \rd x = \frac {-x^{m + 1} \paren {a x + b}^{n + 1} } {\paren {n + 1} b} + \frac {m + n + 2} {\paren {n + 1} b} \int x^m \paren {a x + b}^{n + 1} \rd x$ | From [[Primitive of Power of x by Power of a x + b/Decrement of Power of x|Primitive of Power of $x$ by Power of $a x + b$: Decrement of Power of $x$]]:
:$\ds \int x^m \paren {a x + b}^n \rd x = \frac {x^{m + 1} \paren {a x + b}^n} {m + n + 1} + \frac {n b} {m + n + 1} \int x^m \paren {a x + b}^{n - 1} \rd x$
Substi... | Primitive of Power of x by Power of a x + b/Increment of Power of a x + b/Proof 1 | https://proofwiki.org/wiki/Primitive_of_Power_of_x_by_Power_of_a_x_+_b/Increment_of_Power_of_a_x_+_b | https://proofwiki.org/wiki/Primitive_of_Power_of_x_by_Power_of_a_x_+_b/Increment_of_Power_of_a_x_+_b/Proof_1 | [
"Primitive of Power of x by Power of a x + b"
] | [] | [
"Primitive of Power of x by Power of a x + b/Decrement of Power of x"
] |
proofwiki-9187 | Primitive of Power of x by Power of a x + b/Increment of Power of a x + b | :$\ds \int x^m \paren {a x + b}^n \rd x = \frac {-x^{m + 1} \paren {a x + b}^{n + 1} } {\paren {n + 1} b} + \frac {m + n + 2} {\paren {n + 1} b} \int x^m \paren {a x + b}^{n + 1} \rd x$ | From Primitive of Power of $a x + b$ by Power of $p x + q$: Increment of Power:
:$\ds \int \paren {a x + b}^m \paren {p x + q}^n \rd x = \frac 1 {\paren {n + 1} \paren {b p - a q} } \paren {\paren {a x + b}^{m + 1} \paren {p x + q}^{n + 1} - a \paren {m + n + 2} \int \paren {a x + b}^m \paren {p x + q}^{n + 1} \rd x}$
... | :$\ds \int x^m \paren {a x + b}^n \rd x = \frac {-x^{m + 1} \paren {a x + b}^{n + 1} } {\paren {n + 1} b} + \frac {m + n + 2} {\paren {n + 1} b} \int x^m \paren {a x + b}^{n + 1} \rd x$ | From [[Primitive of Power of a x + b by Power of p x + q/Increment of Power|Primitive of Power of $a x + b$ by Power of $p x + q$: Increment of Power]]:
:$\ds \int \paren {a x + b}^m \paren {p x + q}^n \rd x = \frac 1 {\paren {n + 1} \paren {b p - a q} } \paren {\paren {a x + b}^{m + 1} \paren {p x + q}^{n + 1} - a \pa... | Primitive of Power of x by Power of a x + b/Increment of Power of a x + b/Proof 2 | https://proofwiki.org/wiki/Primitive_of_Power_of_x_by_Power_of_a_x_+_b/Increment_of_Power_of_a_x_+_b | https://proofwiki.org/wiki/Primitive_of_Power_of_x_by_Power_of_a_x_+_b/Increment_of_Power_of_a_x_+_b/Proof_2 | [
"Primitive of Power of x by Power of a x + b"
] | [] | [
"Primitive of Power of a x + b by Power of p x + q/Increment of Power"
] |
proofwiki-9188 | Primitive of Power of x by Power of a x + b/Increment of Power of x | :$\ds \int x^m \paren {a x + b}^n \rd x = \frac {x^{m + 1} \paren {a x + b}^{n + 1} } {\paren {m + 1} b} - \frac {\paren {m + n + 2} a} {\paren {m + 1} b} \int x^{m + 1} \paren {a x + b}^n \rd x$ | From Primitive of Power of $x$ by Power of $a x + b$: Decrement of Power of $x$:
:$\ds \int x^m \paren {a x + b}^n \rd x = \frac {x^m \paren {a x + b}^{n + 1} } {\paren {m + n + 1} a} - \frac {m b} {\paren {m + n + 1} a} \int x^{m - 1} \paren {a x + b}^n \rd x$
Substituting $m + 1$ for $m$:
{{begin-eqn}}
{{eqn | l = \i... | :$\ds \int x^m \paren {a x + b}^n \rd x = \frac {x^{m + 1} \paren {a x + b}^{n + 1} } {\paren {m + 1} b} - \frac {\paren {m + n + 2} a} {\paren {m + 1} b} \int x^{m + 1} \paren {a x + b}^n \rd x$ | From [[Primitive of Power of x by Power of a x + b/Decrement of Power of x|Primitive of Power of $x$ by Power of $a x + b$: Decrement of Power of $x$]]:
:$\ds \int x^m \paren {a x + b}^n \rd x = \frac {x^m \paren {a x + b}^{n + 1} } {\paren {m + n + 1} a} - \frac {m b} {\paren {m + n + 1} a} \int x^{m - 1} \paren {a x... | Primitive of Power of x by Power of a x + b/Increment of Power of x/Proof 1 | https://proofwiki.org/wiki/Primitive_of_Power_of_x_by_Power_of_a_x_+_b/Increment_of_Power_of_x | https://proofwiki.org/wiki/Primitive_of_Power_of_x_by_Power_of_a_x_+_b/Increment_of_Power_of_x/Proof_1 | [
"Primitive of Power of x by Power of a x + b"
] | [] | [
"Primitive of Power of x by Power of a x + b/Decrement of Power of x"
] |
proofwiki-9189 | Primitive of Power of x by Power of a x + b/Increment of Power of x | :$\ds \int x^m \paren {a x + b}^n \rd x = \frac {x^{m + 1} \paren {a x + b}^{n + 1} } {\paren {m + 1} b} - \frac {\paren {m + n + 2} a} {\paren {m + 1} b} \int x^{m + 1} \paren {a x + b}^n \rd x$ | From Primitive of Power of $a x + b$ by Power of $p x + q$: Increment of Power:
:$\ds \int \paren {a x + b}^m \paren {p x + q}^n \rd x = \frac 1 {\paren {n + 1} \paren {b p - a q} } \paren {\paren {a x + b}^{m + 1} \paren {p x + q}^{n + 1} - a \paren {m + n + 2} \int \paren {a x + b}^m \paren {p x + q}^{n + 1} \rd x}$
... | :$\ds \int x^m \paren {a x + b}^n \rd x = \frac {x^{m + 1} \paren {a x + b}^{n + 1} } {\paren {m + 1} b} - \frac {\paren {m + n + 2} a} {\paren {m + 1} b} \int x^{m + 1} \paren {a x + b}^n \rd x$ | From [[Primitive of Power of a x + b by Power of p x + q/Increment of Power|Primitive of Power of $a x + b$ by Power of $p x + q$: Increment of Power]]:
:$\ds \int \paren {a x + b}^m \paren {p x + q}^n \rd x = \frac 1 {\paren {n + 1} \paren {b p - a q} } \paren {\paren {a x + b}^{m + 1} \paren {p x + q}^{n + 1} - a \pa... | Primitive of Power of x by Power of a x + b/Increment of Power of x/Proof 2 | https://proofwiki.org/wiki/Primitive_of_Power_of_x_by_Power_of_a_x_+_b/Increment_of_Power_of_x | https://proofwiki.org/wiki/Primitive_of_Power_of_x_by_Power_of_a_x_+_b/Increment_of_Power_of_x/Proof_2 | [
"Primitive of Power of x by Power of a x + b"
] | [] | [
"Primitive of Power of a x + b by Power of p x + q/Increment of Power"
] |
proofwiki-9190 | Primitive of Reciprocal of Root of a x + b | :$\ds \int \frac {\d x} {\sqrt{a x + b} } = \frac {2 \sqrt {a x + b} } a + C$
where $a x + b > 0$. | Put $u = \sqrt{a x + b}$.
Then:
{{begin-eqn}}
{{eqn | l = \int \frac {\d x} {\sqrt {a x + b} }
| r = \frac 2 a \int u \frac {\d u} u
| c = Primitive of Function of $\sqrt {a x + b}$
}}
{{eqn | r = \frac 2 a \int \rd u
| c =
}}
{{eqn | r = \frac 2 a u + C
| c = Primitive of Constant
}}
{{eqn | r... | :$\ds \int \frac {\d x} {\sqrt{a x + b} } = \frac {2 \sqrt {a x + b} } a + C$
where $a x + b > 0$. | Put $u = \sqrt{a x + b}$.
Then:
{{begin-eqn}}
{{eqn | l = \int \frac {\d x} {\sqrt {a x + b} }
| r = \frac 2 a \int u \frac {\d u} u
| c = [[Primitive of Function of Root of a x + b|Primitive of Function of $\sqrt {a x + b}$]]
}}
{{eqn | r = \frac 2 a \int \rd u
| c =
}}
{{eqn | r = \frac 2 a u + C... | Primitive of Reciprocal of Root of a x + b | https://proofwiki.org/wiki/Primitive_of_Reciprocal_of_Root_of_a_x_+_b | https://proofwiki.org/wiki/Primitive_of_Reciprocal_of_Root_of_a_x_+_b | [
"Primitive of Reciprocal of Root of a x + b",
"Primitives involving Root of a x + b",
"Primitives involving Reciprocals"
] | [] | [
"Primitive of Function of Root of a x + b",
"Primitive of Constant"
] |
proofwiki-9191 | Primitive of Reciprocal of Root of a x + b | :$\ds \int \frac {\d x} {\sqrt{a x + b} } = \frac {2 \sqrt {a x + b} } a + C$
where $a x + b > 0$. | First let us express the integrand in the following form:
{{begin-eqn}}
{{eqn | n = 1
| l = \int \frac {\d x} {\sqrt {\paren {a x + b} \paren {p x + q} } }
| r = \int \frac {\d x} {\sqrt {a p \paren {x - \paren {-\frac b a} } \paren {x - \paren {-\frac q p} } } }
| c =
}}
{{end-eqn}}
Recall the defin... | :$\ds \int \frac {\d x} {\sqrt{a x + b} } = \frac {2 \sqrt {a x + b} } a + C$
where $a x + b > 0$. | First let us express the [[Definition:Integrand|integrand]] in the following form:
{{begin-eqn}}
{{eqn | n = 1
| l = \int \frac {\d x} {\sqrt {\paren {a x + b} \paren {p x + q} } }
| r = \int \frac {\d x} {\sqrt {a p \paren {x - \paren {-\frac b a} } \paren {x - \paren {-\frac q p} } } }
| c =
}}
{{... | Primitive of Reciprocal of Root of a x + b by Root of p x + q/a p less than 0/Proof 2 | https://proofwiki.org/wiki/Primitive_of_Reciprocal_of_Root_of_a_x_+_b | https://proofwiki.org/wiki/Primitive_of_Reciprocal_of_Root_of_a_x_+_b_by_Root_of_p_x_+_q/a_p_less_than_0/Proof_2 | [
"Primitive of Reciprocal of Root of a x + b",
"Primitives involving Root of a x + b",
"Primitives involving Reciprocals"
] | [] | [
"Definition:Integration/Integrand",
"Definition:Euler Substitution/Third",
"Definition:Fraction/Numerator",
"Definition:Fraction/Denominator",
"Integration by Substitution",
"Primitive of Reciprocal of x squared plus a squared/Arctangent Form",
"Arccotangent of Reciprocal equals Arctangent",
"Sum of A... |
proofwiki-9192 | Primitive of x over Root of a x + b | :$\ds \int \frac {x \rd x} {\sqrt {a x + b} } = \frac {2 \paren {a x - 2 b} \sqrt {a x + b} } {3 a^2}$ | Let:
{{begin-eqn}}
{{eqn | l = u
| r = \sqrt {a x + b}
| c =
}}
{{eqn | l = x
| r = \frac {u^2 - b} a
| c =
}}
{{end-eqn}}
Thus:
{{begin-eqn}}
{{eqn | l = \map F {\sqrt {a x + b} }
| r = \frac x {\sqrt {a x + b} }
| c =
}}
{{eqn | ll= \leadsto
| l = \map F u
| r = \par... | :$\ds \int \frac {x \rd x} {\sqrt {a x + b} } = \frac {2 \paren {a x - 2 b} \sqrt {a x + b} } {3 a^2}$ | Let:
{{begin-eqn}}
{{eqn | l = u
| r = \sqrt {a x + b}
| c =
}}
{{eqn | l = x
| r = \frac {u^2 - b} a
| c =
}}
{{end-eqn}}
Thus:
{{begin-eqn}}
{{eqn | l = \map F {\sqrt {a x + b} }
| r = \frac x {\sqrt {a x + b} }
| c =
}}
{{eqn | ll= \leadsto
| l = \map F u
| r = ... | Primitive of x over Root of a x + b | https://proofwiki.org/wiki/Primitive_of_x_over_Root_of_a_x_+_b | https://proofwiki.org/wiki/Primitive_of_x_over_Root_of_a_x_+_b | [
"Primitives involving Root of a x + b"
] | [] | [
"Primitive of Function of Root of a x + b",
"Primitive of Constant Multiple of Function",
"Primitive of Power",
"Primitive of Constant"
] |
proofwiki-9193 | Primitive of x squared over Root of a x + b | :$\ds \int \frac {x^2 \rd x} {\sqrt {a x + b} } = \frac {2 \paren {3 a^2 x^2 - 4 a b x + 8 b^2} \sqrt {a x + b} } {15 a^3}$ | Let:
{{begin-eqn}}
{{eqn | l = u
| r = \sqrt {a x + b}
| c =
}}
{{eqn | ll= \leadsto
| l = x
| r = \frac {u^2 - b} a
| c =
}}
{{end-eqn}}
Thus:
{{begin-eqn}}
{{eqn | l = \map F {\sqrt {a x + b} }
| r = \frac {x^2} {\sqrt {a x + b} }
| c =
}}
{{eqn | ll= \leadsto
| l = ... | :$\ds \int \frac {x^2 \rd x} {\sqrt {a x + b} } = \frac {2 \paren {3 a^2 x^2 - 4 a b x + 8 b^2} \sqrt {a x + b} } {15 a^3}$ | Let:
{{begin-eqn}}
{{eqn | l = u
| r = \sqrt {a x + b}
| c =
}}
{{eqn | ll= \leadsto
| l = x
| r = \frac {u^2 - b} a
| c =
}}
{{end-eqn}}
Thus:
{{begin-eqn}}
{{eqn | l = \map F {\sqrt {a x + b} }
| r = \frac {x^2} {\sqrt {a x + b} }
| c =
}}
{{eqn | ll= \leadsto
| ... | Primitive of x squared over Root of a x + b | https://proofwiki.org/wiki/Primitive_of_x_squared_over_Root_of_a_x_+_b | https://proofwiki.org/wiki/Primitive_of_x_squared_over_Root_of_a_x_+_b | [
"Primitives involving Root of a x + b"
] | [] | [
"Primitive of Function of Root of a x + b",
"Primitive of Constant Multiple of Function",
"Primitive of Power",
"Primitive of Constant"
] |
proofwiki-9194 | Primitive of Reciprocal of x squared minus a squared/Inverse Hyperbolic Cotangent Form | Let $a \in \R_{>0}$ be a strictly positive real constant.
Let $\size x > a$.
Then:
:$\ds \int \frac {\d x} {x^2 - a^2} = -\frac 1 a \coth^{-1} {\frac x a} + C$ | Let $\size x > a$.
Let:
{{begin-eqn}}
{{eqn | l = u
| r = \coth^{-1} {\frac x a}
| c = {{Defof|Real Inverse Hyperbolic Cotangent}}, which is defined where $\size {\dfrac x a} > 1$
}}
{{eqn | ll= \leadsto
| l = x
| r = a \coth u
| c =
}}
{{eqn | ll= \leadsto
| l = \frac {\d x} {\d u}... | Let $a \in \R_{>0}$ be a [[Definition:Strictly Positive Real Number|strictly positive real]] [[Definition:Constant|constant]].
Let $\size x > a$.
Then:
:$\ds \int \frac {\d x} {x^2 - a^2} = -\frac 1 a \coth^{-1} {\frac x a} + C$ | Let $\size x > a$.
Let:
{{begin-eqn}}
{{eqn | l = u
| r = \coth^{-1} {\frac x a}
| c = {{Defof|Real Inverse Hyperbolic Cotangent}}, which is defined where $\size {\dfrac x a} > 1$
}}
{{eqn | ll= \leadsto
| l = x
| r = a \coth u
| c =
}}
{{eqn | ll= \leadsto
| l = \frac {\d x} {\d ... | Primitive of Reciprocal of x squared minus a squared/Inverse Hyperbolic Cotangent Form/Proof | https://proofwiki.org/wiki/Primitive_of_Reciprocal_of_x_squared_minus_a_squared/Inverse_Hyperbolic_Cotangent_Form | https://proofwiki.org/wiki/Primitive_of_Reciprocal_of_x_squared_minus_a_squared/Inverse_Hyperbolic_Cotangent_Form/Proof | [
"Primitive of Reciprocal of x squared minus a squared"
] | [
"Definition:Strictly Positive/Real Number",
"Definition:Constant"
] | [
"Derivative of Hyperbolic Cotangent",
"Integration by Substitution",
"Primitive of Constant Multiple of Function",
"Difference of Squares of Hyperbolic Cotangent and Cosecant",
"Integral of Constant"
] |
proofwiki-9195 | Primitive of Reciprocal of x by Root of a x + b | For $a > 0$ and for $x \ne 0$:
:<nowiki>$\ds \int \frac {\d x} {x \sqrt {a x + b} } = \begin {cases}
\dfrac 1 {\sqrt b} \ln \size {\dfrac {\sqrt {a x + b} - \sqrt b} {\sqrt {a x + b} + \sqrt b} } + C & : b > 0 \\ \\
\dfrac 2 {\sqrt {-b} } \arctan \sqrt {\dfrac {a x + b} {-b} } + C & : b < 0 \end {cases}$</nowiki>
where... | Let:
{{begin-eqn}}
{{eqn | l = u
| r = \sqrt {a x + b}
| c =
}}
{{eqn | ll= \leadsto
| l = x
| r = \frac {u^2 - b} a
| c =
}}
{{end-eqn}}
Thus:
{{begin-eqn}}
{{eqn | l = \map F {\sqrt {a x + b} }
| r = \frac 1 {x \sqrt {a x + b} }
| c =
}}
{{eqn | ll= \leadsto
| l = \m... | For $a > 0$ and for $x \ne 0$:
:<nowiki>$\ds \int \frac {\d x} {x \sqrt {a x + b} } = \begin {cases}
\dfrac 1 {\sqrt b} \ln \size {\dfrac {\sqrt {a x + b} - \sqrt b} {\sqrt {a x + b} + \sqrt b} } + C & : b > 0 \\ \\
\dfrac 2 {\sqrt {-b} } \arctan \sqrt {\dfrac {a x + b} {-b} } + C & : b < 0 \end {cases}$</nowiki>
where... | Let:
{{begin-eqn}}
{{eqn | l = u
| r = \sqrt {a x + b}
| c =
}}
{{eqn | ll= \leadsto
| l = x
| r = \frac {u^2 - b} a
| c =
}}
{{end-eqn}}
Thus:
{{begin-eqn}}
{{eqn | l = \map F {\sqrt {a x + b} }
| r = \frac 1 {x \sqrt {a x + b} }
| c =
}}
{{eqn | ll= \leadsto
| l =... | Primitive of Reciprocal of x by Root of a x + b | https://proofwiki.org/wiki/Primitive_of_Reciprocal_of_x_by_Root_of_a_x_+_b | https://proofwiki.org/wiki/Primitive_of_Reciprocal_of_x_by_Root_of_a_x_+_b | [
"Primitives involving Root of a x + b"
] | [] | [
"Primitive of Function of Root of a x + b",
"Primitive of Constant Multiple of Function",
"Primitive of Reciprocal of x squared minus a squared/Logarithm Form",
"Primitive of Reciprocal of x squared plus a squared/Arctangent Form"
] |
proofwiki-9196 | Primitive of Reciprocal of x squared by Root of a x + b | :$\ds \int \frac {\d x} {x^2 \sqrt {a x + b} } = -\frac {\sqrt {a x + b} } {b x} - \frac a {2 b} \int \frac {\d x} {x \sqrt {a x + b} }$ | From Primitive of Power of $x$ by Power of $a x + b$: Increment of Power of $x$:
:$\ds \int x^m \paren {a x + b}^n \rd x = \frac {x^{m + 1} \paren {a x + b}^{n + 1} } {\paren {m + 1} b} - \frac {\paren {m + n + 2} a} {\paren {m + 1} b} \int x^{m + 1} \paren {a x + b}^n \rd x$
Setting $m = -2$ and $n = -\dfrac 1 2$:
{{b... | :$\ds \int \frac {\d x} {x^2 \sqrt {a x + b} } = -\frac {\sqrt {a x + b} } {b x} - \frac a {2 b} \int \frac {\d x} {x \sqrt {a x + b} }$ | From [[Primitive of Power of x by Power of a x + b/Increment of Power of x|Primitive of Power of $x$ by Power of $a x + b$: Increment of Power of $x$]]:
:$\ds \int x^m \paren {a x + b}^n \rd x = \frac {x^{m + 1} \paren {a x + b}^{n + 1} } {\paren {m + 1} b} - \frac {\paren {m + n + 2} a} {\paren {m + 1} b} \int x^{m +... | Primitive of Reciprocal of x squared by Root of a x + b | https://proofwiki.org/wiki/Primitive_of_Reciprocal_of_x_squared_by_Root_of_a_x_+_b | https://proofwiki.org/wiki/Primitive_of_Reciprocal_of_x_squared_by_Root_of_a_x_+_b | [
"Primitives involving Root of a x + b",
"Primitives involving Reciprocals"
] | [] | [
"Primitive of Power of x by Power of a x + b/Increment of Power of x"
] |
proofwiki-9197 | Primitive of Root of a x + b | :$\ds \int \sqrt {a x + b} \rd x = \frac {2 \sqrt {\paren {a x + b}^3} } {3 a}$ | Let $u = \sqrt{a x + b}$.
Then:
{{begin-eqn}}
{{eqn | l = \int \sqrt {a x + b} \rd x
| r = \frac 2 a \int u^2 \rd u
| c = Primitive of Function of $\sqrt {a x + b}$
}}
{{eqn | r = \frac 2 a \frac {u^3} 3
| c = Primitive of Power
}}
{{eqn | r = \frac {2 \sqrt {\paren {a x + b}^3} } {3 a}
| c = su... | :$\ds \int \sqrt {a x + b} \rd x = \frac {2 \sqrt {\paren {a x + b}^3} } {3 a}$ | Let $u = \sqrt{a x + b}$.
Then:
{{begin-eqn}}
{{eqn | l = \int \sqrt {a x + b} \rd x
| r = \frac 2 a \int u^2 \rd u
| c = [[Primitive of Function of Root of a x + b|Primitive of Function of $\sqrt {a x + b}$]]
}}
{{eqn | r = \frac 2 a \frac {u^3} 3
| c = [[Primitive of Power]]
}}
{{eqn | r = \frac {2... | Primitive of Root of a x + b | https://proofwiki.org/wiki/Primitive_of_Root_of_a_x_+_b | https://proofwiki.org/wiki/Primitive_of_Root_of_a_x_+_b | [
"Primitives involving Root of a x + b"
] | [] | [
"Primitive of Function of Root of a x + b",
"Primitive of Power"
] |
proofwiki-9198 | Primitive of Root of a x + b | :$\ds \int \sqrt {a x + b} \rd x = \frac {2 \sqrt {\paren {a x + b}^3} } {3 a}$ | From Primitive of Power of $x$ by Power of $a x + b$: Decrement of Power of $a x + b$:
:$\ds \int x^m \paren {a x + b}^n \rd x = \frac {x^{m + 1} \paren {a x + b}^n} {m + n + 1} + \frac {n b} {m + n + 1} \int x^m \paren {a x + b}^{n - 1} \rd x$
Putting $m = -1$ and $n = \dfrac 1 2$:
{{begin-eqn}}
{{eqn | l = \int \frac... | :$\ds \int \sqrt {a x + b} \rd x = \frac {2 \sqrt {\paren {a x + b}^3} } {3 a}$ | From [[Primitive of Power of x by Power of a x + b/Decrement of Power of a x + b|Primitive of Power of $x$ by Power of $a x + b$: Decrement of Power of $a x + b$]]:
:$\ds \int x^m \paren {a x + b}^n \rd x = \frac {x^{m + 1} \paren {a x + b}^n} {m + n + 1} + \frac {n b} {m + n + 1} \int x^m \paren {a x + b}^{n - 1} \rd... | Primitive of Root of a x + b over x/Proof 1 | https://proofwiki.org/wiki/Primitive_of_Root_of_a_x_+_b | https://proofwiki.org/wiki/Primitive_of_Root_of_a_x_+_b_over_x/Proof_1 | [
"Primitives involving Root of a x + b"
] | [] | [
"Primitive of Power of x by Power of a x + b/Decrement of Power of a x + b"
] |
proofwiki-9199 | Primitive of Root of a x + b | :$\ds \int \sqrt {a x + b} \rd x = \frac {2 \sqrt {\paren {a x + b}^3} } {3 a}$ | Let:
{{begin-eqn}}
{{eqn | l = v
| r = \sqrt x
| c =
}}
{{eqn | ll= \leadsto
| l = \frac {\d v} {\d x}
| r = \frac 1 {2 \sqrt x}
| c = Power Rule for Derivatives
}}
{{eqn | l = u
| r = \frac {2 \sqrt {a x + b} } {\sqrt x}
| c =
}}
{{eqn | ll= \leadsto
| l = \frac {\d u}... | :$\ds \int \sqrt {a x + b} \rd x = \frac {2 \sqrt {\paren {a x + b}^3} } {3 a}$ | Let:
{{begin-eqn}}
{{eqn | l = v
| r = \sqrt x
| c =
}}
{{eqn | ll= \leadsto
| l = \frac {\d v} {\d x}
| r = \frac 1 {2 \sqrt x}
| c = [[Power Rule for Derivatives]]
}}
{{eqn | l = u
| r = \frac {2 \sqrt {a x + b} } {\sqrt x}
| c =
}}
{{eqn | ll= \leadsto
| l = \frac {\... | Primitive of Root of a x + b over x/Proof 2 | https://proofwiki.org/wiki/Primitive_of_Root_of_a_x_+_b | https://proofwiki.org/wiki/Primitive_of_Root_of_a_x_+_b_over_x/Proof_2 | [
"Primitives involving Root of a x + b"
] | [] | [
"Power Rule for Derivatives",
"Quotient Rule for Derivatives",
"Integration by Parts"
] |
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