nyuuzyou/wikis huggingface dataset:
Primitive of Function on Connected Domain

== Theorem ==
Let $f: D \to \C$ be a continuous complex function, where $D$ is a connected domain.
Then the following three conditions are equivalent:
:$(1): \quad$ $f$ has a primitive.
:$(2): \quad$ For any two contours $C_1, C_2$ in $D$ with identical start points $z_1 \in D$ and end points $z_2 \in D$, we have:
::$\ds \int_{C_1} \map f z \rd z = \int_{C_2} \map f z \rd z$
:$(3): \quad$ For all closed contours $C$ in $D$, we have:
::$\ds \oint_C \map f z \rd z = 0$
If the conditions hold, we can choose any $z_0 \in D$ and define a primitive $F: D \to \C$ of $f$ by:
:$\ds \map F w = \int_{C_w} \map f z \rd z$
where $C_w$ is any contour in $D$ with start point $z_0$ and end point $w$.
== Proof ==
=== $(1)$ implies $(2)$ ===
If $F$ is a primitive of $f$, we have:
=== $(2)$ implies $(3)$ ===
Let $C$ be a closed contour in $D$ with endpoints $z_0$.
Let the constant function $\gamma: \closedint 0 1 \to D$ with $\map \gamma t = z_0$ be the parameterization of a contour $C_0$.
Then:
=== $(3)$ implies $(1)$ ===
This follows from Zero Staircase Integral Condition for Primitive.
=== Construction of a Primitive ===
If the conditions hold, we choose $z_0 \in D$ and define a function $F: D \to \C$ of $f$ by:
:$\ds \map F w = \int_{C_w} \map f z \rd z$
where $C_w$ is any contour in $D$ with start point $z_0$ and end point $w$.
From Connected Domain is Connected by Staircase Contours, it follows that we can choose $C_w$ to be a staircase contour.
If $C_w'$ is another contour in $D$ with the same endpoints as $C_w$, then:
:$\ds \int_{C_w'} \map f z \rd z = \int_{C_w} \map f z \rd z$
by condition $(2)$, so $F$ is well-defined.
From Zero Staircase Integral Condition for Primitive, it follows that $F$ is a primitive of $f$.
== Sources ==
* : $\S 2.3$
Category:Complex Analysis
Primitive of Function under its Derivative

== Theorem ==
Let $f$ be a real function which is integrable.
Then:
:$\ds \int \frac {\map {f'} x} {\map f x} \rd x = \ln \size {\map f x} + C$
where $C$ is an arbitrary constant.
== Proof ==
By Integration by Substitution (with appropriate renaming of variables):
:$\ds \int \map g u \rd u = \int \map g {\map f x} \map {f'} x \rd x$
Then:
== Also presented as ==
== Examples ==
== Sources ==
* : Chapter $\text {II}$: Integration of Elementary Functions: $\S 8$. Change of Variable
* : $\text {II}$. Calculus: Integration
* : Chapter $10$: Integration: $10.4$. Standard integrals: General Rules: $\text {III}$.
* : Appendix $\text I$: Table of Indefinite Integrals $7$.
* : Appendix: Table $2$: Integrals
* : Appendix: Table $2$: Integrals
Category:Primitive of Function under its Derivative
Category:Primitives
Primitive of Function under its Derivative/Also presented as

== Primitive of Function under its Derivative: Also presented as == 
Primitive of Function under its Derivative can also be presented as:
:$\ds \int \frac {\d u} u = \ln \size u + C$
where it is understood that $u$ is a function of $x$.
== Sources ==
* : Chapter $\text {II}$: Integration of Elementary Functions: $\S 8$. Change of Variable
* : $\S 14$: General Rules of Integration: $14.8$
* : $\S 16$: Indefinite Integrals: General Rules of Integration: $16.8.$
Category:Primitive of Function under its Derivative
Primitive of Function under its Derivative/Examples

== Examples of Use of Primitive of Function under its Derivative ==
=== Primitive of $\dfrac {2 x - 5} {\paren {x - 2} \paren {x - 3} }$ ===
=== Primitive of $\dfrac {2 x + a} {x^2 + a x + b}$ ===
Category:Examples of Use of Primitive of Function under its Derivative
Primitive of Function under its Derivative/Examples/2 x + a over x^2 + a x + b

== Example of Use of Primitive of Function under its Derivative ==
:$\ds \int \dfrac {2 x + a} {x^2 + a x + b} \rd x = \ln \size {x^2 + a x + b} + C$
== Proof ==
== Sources ==
* : $6$. Integral Calculus: Appendix: Table of Fundamental Standard Integrals
Category:Examples of Use of Primitive of Function under its Derivative
Primitive of Function under its Derivative/Examples/2 x - 5 over (x - 2) (x - 3)

== Example of Use of Primitive of Function under its Derivative ==
:$\ds \int \dfrac {2 x - 5} {\paren {x - 2} \paren {x - 3} } = \ln \size {\paren {x - 2} \paren {x - 3} } + C$
== Proof ==
== Sources ==
* : $\text {II}$. Calculus: Exercises $\text {XIV}$: $22$.
Category:Examples of Use of Primitive of Function under its Derivative
Primitive of Function under its Derivative/Mistake

== Source Works ==
:
:$\text {II}$. Calculus: Integration
:
:Chapter $10$: Integration: $10.4$. Standard integrals: General Rules: $\text {III}$
== Mistake ==
:$\ds \int \frac {\map {f'} x} {\map f x} \rd x = \map \ln {\map f x}$
== Correction ==
It should be:
:$\ds \int \frac {\map {f'} x} {\map f x} \rd x = \ln \size {\map f x} + C$
from Primitive of Function under its Derivative.
== Sources ==
* : $\text {II}$. Calculus: Integration
* : Chapter $10$: Integration: $10.4$. Standard integrals: General Rules: $\text {III}$.
Category:Mistakes/A Note Book in Pure Mathematics
Category:Mistakes/A Course in Pure Mathematics
Primitive of General Exponential of a x

== Theorem ==
:$\ds \int b^{a x} \rd x = \frac {b^{a x} } {a \ln b} + C$
where:
:$a \ne 0$
:$b > 0, b \ne 1$
== Proof ==
== Sources ==
* : Back endpapers: A Brief Table of Integrals: $103$.
Category:Primitive of General Exponential of a x
Category:Primitives involving Exponential Function
Primitive of Half Integer Power of a x squared plus b x plus c

== Theorem ==
Let $a \in \R_{\ne 0}$.
Then:
:$\ds \int \paren {a x^2 + b x + c}^{n + \frac 1 2} \rd x = \frac {\paren {2 a x + b} \paren {a x^2 + b x + c}^{n + \frac 1 2} } {4 a \paren {n + 1} } + \frac {\paren {2 n + 1} \paren {4 a c - b^2} } {8 a \paren {n + 1} } \int \paren {a x^2 + b x + c}^{n - \frac 1 2} \rd x$ 
== Proof ==
Let:
Also let $q = 4 a c - b^2$.
Then:
From Primitive of $\dfrac {\paren {p x + q}^n} {\sqrt {a x + b} }$:
:$\ds \int \frac {\paren {p x + q}^n} {\sqrt {a x + b} } \rd x = \frac {2 \paren {p x + q}^n \sqrt {a x + b} } {\paren {2 n + 1} a} + \frac {2 n \paren {a q - b p} } {\paren {2 n + 1} a} \int \frac {\paren {p x + q}^{n - 1} } {\sqrt {a x + b} } \rd x$
Here $p = 1, a = 1, b = 0$ and $n := n + \dfrac 1 2$:
== Sources ==
* : $\S 14$: Integrals involving $\sqrt {a x^2 + b x + c}$: $14.295$
Category:Primitives involving Root of a x squared plus b x plus c
Primitive of Hyperbolic Cosecant Function

== Theorem ==
=== Logarithm Form ===
=== Hyperbolic Tangent Form ===
=== Inverse Hyperbolic Cotangent Form ===
=== Inverse Hyperbolic Cotangent of Hyperbolic Cosine Form ===
== Also see ==
* Primitive of Hyperbolic Sine Function
* Primitive of Hyperbolic Cosine Function
* Primitive of Hyperbolic Tangent Function
* Primitive of Hyperbolic Cotangent Function
* Primitive of Hyperbolic Secant Function
== Sources ==
* : $\S 14$: General Rules of Integration: $14.30$
* : $\S 16$: Indefinite Integrals: General Rules of Integration: $16.30.$
Category:Primitive of Hyperbolic Cosecant Function
Category:Primitives of Hyperbolic Functions
Category:Hyperbolic Cosecant Function
Category:Primitives involving Hyperbolic Cosecant Function
Primitive of Hyperbolic Cosecant Function/Hyperbolic Tangent Form

== Theorem ==
:$\ds \int \csch x \rd x = \ln \size {\tanh \frac x 2} + C$
where $\tanh \dfrac x 2 \ne 0$.
== Proof 1 ==
== Proof 2 ==
== Sources ==
* : Chapter $10$: Integration: $10.4$. Standard integrals: Other Standard Results: $\text {(xxviii)}$
* : $\S 14$: General Rules of Integration: $14.30$
* : Appendix $2$: Table of derivatives and integrals of common functions: Hyperbolic functions
* : Appendix: Table $2$: Integrals
* : Appendix: Table $2$: Integrals
* : $\S 16$: Indefinite Integrals: General Rules of Integration: $16.30.$
Category:Primitive of Hyperbolic Cosecant Function
Primitive of Hyperbolic Cosecant Function/Inverse Hyperbolic Cotangent Form

== Theorem ==
:$\ds \int \csch x \rd x = -2 \map {\coth^{-1} } {e^x} + C$
== Proof ==
Let:
Let:
Then:
== Also see ==
* Primitive of Hyperbolic Secant Function
== Sources ==
* : $\S 14$: Indefinite Integrals: General Rules of Integration: $14.30$
::(in which a mistake apppears)
* : $\S 16$: Indefinite Integrals: General Rules of Integration: $16.30.$
::(in which the same mistake apppears)
Category:Primitive of Hyperbolic Cosecant Function
Category:Primitives involving Hyperbolic Cotangent Function
Primitive of Hyperbolic Cosecant Function/Inverse Hyperbolic Cotangent Form/Mistake

== Source Work ==
:Chapter $14$: Indefinite Integrals
::General Rules of Integration: $14.30$
:Chapter $16$: Indefinite Integrals
::General Rules of Integration: $16.30.$
== Mistake ==
:$\ds \int \csch u \rd u = -\map {\coth^{-1} } {e^u} + C$
== Correction ==
As demonstrated in Primitive of Hyperbolic Cosecant Function: Inverse Hyperbolic Cotangent Form the correct expression is in fact:
:$\ds \int \csch u \rd u = -2 \map {\coth^{-1} } {e^u} + C$
== Sources ==
* : $\S 14$: Indefinite Integrals: General Rules of Integration: $14.30$
* : $\S 16$: Indefinite Integrals: General Rules of Integration: $16.30.$
Category:Mistakes/Mathematical Handbook of Formulas and Tables
Primitive of Hyperbolic Cosecant Function/Inverse Hyperbolic Cotangent of Hyperbolic Cosine Form

== Theorem ==
:$\ds \int \csch x \rd x = -\map {\coth^{-1} } {\cosh x} + C$
== Proof 1 ==
== Proof 2 ==
== Also see ==
* Primitive of Hyperbolic Secant Function
== Sources ==
* : Chapter $5$. Integrals: Integrals of Special Functions: $20$
Category:Primitive of Hyperbolic Cosecant Function
Category:Primitives involving Hyperbolic Cotangent Function
Primitive of Hyperbolic Cosecant Function/Logarithm Form

== Theorem ==
:$\ds \int \csch x \rd x = -\ln \size {\csch x + \coth x} + C$
where $\csch x + \coth x \ne 0$.
== Proof ==
Let:
Then:
== Also see ==
* Primitive of Hyperbolic Secant Function
Category:Primitive of Hyperbolic Cosecant Function
Primitive of Hyperbolic Cosecant of a x

== Theorem ==
:$\ds \int \csch a x \rd x = \frac 1 a \ln \size {\tanh \frac {a x} 2} + C$
== Proof ==
== Also see ==
* Primitive of $\sinh a x$
* Primitive of $\cosh a x$
* Primitive of $\tanh a x$
* Primitive of $\coth a x$
* Primitive of $\sech a x$
== Sources ==
* : $\S 14$: Integrals involving $\csch a x$: $14.636$
* : Back endpapers: A Brief Table of Integrals: $130$.
* : $\S 17$: Tables of Special Indefinite Integrals: $(32)$ Integrals Involving $\csch a x$: $17.33.1.$
Category:Primitives involving Hyperbolic Cosecant Function
Primitive of Hyperbolic Cosecant of a x over x

== Theorem ==
where $B_{2 k}$ denotes the $2 k$th Bernoulli number.
== Proof ==
== Also see ==
* Primitive of $\dfrac {\sinh a x} x$
* Primitive of $\dfrac {\cosh a x} x$
* Primitive of $\dfrac {\tanh a x} x$
* Primitive of $\dfrac {\coth a x} x$
* Primitive of $\dfrac {\sech a x} x$
== Sources ==
* : $\S 14$: Integrals involving $\csch a x$: $14.643$
* : $\S 17$: Tables of Special Indefinite Integrals: $(33)$ Integrals Involving $\csch a x$: $17.33.6.$
Category:Primitives involving Hyperbolic Cosecant Function
Primitive of Hyperbolic Cosine Function

== Theorem ==
:$\ds \int \cosh x \rd x = \sinh x + C$
where $C$ is an arbitrary constant.
== Proof ==
From Derivative of Hyperbolic Sine:
:$\map {\dfrac \d {\d x} } {\sinh x} = \cosh x$
The result follows from the definition of primitive.
== Also see ==
* Primitive of Hyperbolic Sine Function
* Primitive of Hyperbolic Tangent Function
* Primitive of Hyperbolic Cotangent Function
* Primitive of Hyperbolic Secant Function
* Primitive of Hyperbolic Cosecant Function
== Sources ==
* : Chapter $\text {II}$: Integration of Elementary Functions: $\S 7$. Standard Integrals: $8$.
* : Chapter $\text {III}$: Integration: Integration
* : Chapter $10$: Integration: $10.4$. Standard integrals: Other Standard Results: $\text {(xxiv)}$
* : $\S 14$: General Rules of Integration: $14.26$
* : Chapter $5$. Integrals: Integrals of Special Functions: $16$
* : $6$. Integral Calculus: Appendix: Table of Fundamental Standard Integrals
* 
* : Appendix $2$: Table of derivatives and integrals of common functions: Hyperbolic functions
* : Appendix: Table $2$: Integrals
* : Appendix: Table $2$: Integrals
* : $\S 16$: Indefinite Integrals: General Rules of Integration: $16.26.$
* : Appendix $7$: Integrals
* : Appendix $8$: Integrals
Category:Primitives of Hyperbolic Functions
Category:Hyperbolic Cosine Function
Category:Primitives involving Hyperbolic Cosine Function
Primitive of Hyperbolic Cosine of a x

== Theorem ==
:$\ds \int \cosh a x \rd x = \frac {\sinh a x} a + C$
== Proof ==
== Also see ==
* Primitive of $\sinh a x$
* Primitive of $\tanh a x$
* Primitive of $\coth a x$
* Primitive of $\sech a x$
* Primitive of $\csch a x$
== Sources ==
* : $\S 14$: Integrals involving $\cosh a x$: $14.562$
* : Back endpapers: A Brief Table of Integrals: $114$.
* : $\S 17$: Tables of Special Indefinite Integrals: $(28)$ Integrals Involving $\cosh a x$: $17.28.1.$
Category:Primitives involving Hyperbolic Cosine Function
Primitive of Hyperbolic Cosine of a x by Cosine of p x

== Theorem ==
:$\ds \int \cosh a x \cos p x \rd x = \frac {a \sinh a x \cos p x + p \cosh a x \sin p x} {a^2 + p^2} + C$
== Proof ==
== Also see ==
* Primitive of $\cosh a x \sin p x$
* Primitive of $\sinh a x \sin p x$
* Primitive of $\sinh a x \cos p x$
== Sources ==
* : $\S 14$: Integrals involving $\cosh a x$: $14.574$
Category:Primitives involving Hyperbolic Cosine Function
Category:Primitives involving Cosine Function
Primitive of Hyperbolic Cosine of a x by Hyperbolic Cosine of p x

== Theorem ==
:$\ds \int \cosh a x \cosh p x \rd x = \frac {\map \sinh {a + p} x} {2 \paren {a + p} } + \frac {\map \sinh {a - p} x} {2 \paren {a - p} } + C$
== Proof ==
== Also see ==
* Primitive of $\sinh a x \sinh p x$
* Primitive of $\sinh a x \cosh a x$
* Primitive of $\sinh p x \cosh q x$
== Sources ==
* : $\S 14$: Integrals involving $\cosh a x$: $14.572$
* : $\S 17$: Tables of Special Indefinite Integrals: $(28)$ Integrals Involving $\cosh a x$: $17.28.11.$
Category:Primitives involving Hyperbolic Cosine Function
Primitive of Hyperbolic Cosine of a x by Sine of p x

== Theorem ==
:$\ds \int \cosh a x \sin p x \rd x = \frac {a \sinh a x \sin p x - p \cosh a x \cos p x} {a^2 + p^2} + C$
== Proof ==
== Also see ==
* Primitive of $\cosh a x \cos p x$
* Primitive of $\sinh a x \sin p x$
* Primitive of $\sinh a x \cos p x$
== Sources ==
* : $\S 14$: Integrals involving $\cosh a x$: $14.573$
Category:Primitives involving Hyperbolic Cosine Function
Category:Primitives involving Sine Function
Primitive of Hyperbolic Cosine of a x over Power of x

== Theorem ==
:$\ds \int \frac {\cosh a x \rd x} {x^n} = \frac {-\cosh a x} {\paren {n - 1} x^{n - 1} } + \frac a {n - 1} \int \frac {\sinh a x \rd x} {x^{n - 1} } + C$
== Proof ==
With a view to expressing the problem in the form:
:$\ds \int u \frac {\d v} {\d x} \rd x = u v - \int v \frac {\d u} {\d x} \rd x$
let:
and let:
Then:
== Also see ==
* Primitive of $\dfrac {\sinh a x} {x^n}$
== Sources ==
* : $\S 14$: Integrals involving $\cosh a x$: $14.587$
* : $\S 17$: Tables of Special Indefinite Integrals: $(28)$ Integrals Involving $\cosh a x$: $17.28.14.$
Category:Primitives involving Hyperbolic Cosine Function
Primitive of Hyperbolic Cosine of a x over x

== Theorem ==
== Proof ==
== Also see ==
* Primitive of $\dfrac {\sinh a x} x$
* Primitive of $\dfrac {\tanh a x} x$
* Primitive of $\dfrac {\coth a x} x$
* Primitive of $\dfrac {\sech a x} x$
* Primitive of $\dfrac {\csch a x} x$
== Sources ==
* : $\S 14$: Integrals involving $\cosh a x$: $14.565$
Category:Primitives involving Hyperbolic Cosine Function
Primitive of Hyperbolic Cosine of a x over x squared

== Theorem ==
:$\ds \int \frac {\cosh a x \rd x} {x^2} = -\frac {\cosh a x} x + a \int \frac {\sinh a x \rd x} x$
== Proof ==
With a view to expressing the primitive in the form:
:$\ds \int u \frac {\d v} {\d x} \rd x = u v - \int v \frac {\d u} {\d x} \rd x$
let:
and let:
Then:
== Also see ==
* Primitive of $\dfrac {\sinh a x} x$
* Primitive of $\dfrac {\sinh a x} {x^2}$
== Sources ==
* : $\S 14$: Integrals involving $\cosh a x$: $14.566$
Category:Primitives involving Hyperbolic Cosine Function
Primitive of Hyperbolic Cotangent Function

== Theorem ==
:$\ds \int \coth x \rd x = \ln \size {\sinh x} + C$
where $\sinh x \ne 0$.
== Proof ==
== Also see ==
* Primitive of Hyperbolic Sine Function
* Primitive of Hyperbolic Cosine Function
* Primitive of Hyperbolic Tangent Function
* Primitive of Hyperbolic Secant Function
* Primitive of Hyperbolic Cosecant Function
== Sources ==
* : Chapter $\text {II}$: Integration of Elementary Functions: $\S 8$. Change of Variable
* : Chapter $10$: Integration: $10.4$. Standard integrals: Other Standard Results: $\text {(xxvi)}$
* : $\S 14$: General Rules of Integration: $14.28$
* : Chapter $5$. Integrals: Integrals of Special Functions: $18$
* : $6$. Integral Calculus: Appendix: Table of Fundamental Standard Integrals
* : Appendix $2$: Table of derivatives and integrals of common functions: Hyperbolic functions
* : Appendix: Table $2$: Integrals
* : Appendix: Table $2$: Integrals
* : $\S 16$: Indefinite Integrals: General Rules of Integration: $16.28.$
Category:Primitives of Hyperbolic Functions
Category:Hyperbolic Cotangent Function
Category:Primitives involving Hyperbolic Cotangent Function
Primitive of Hyperbolic Cotangent of a x

== Theorem ==
:$\ds \int \coth a x \rd x = \frac {\ln \size {\sinh a x} } a + C$
== Proof ==
== Also see ==
* Primitive of $\sinh a x$
* Primitive of $\cosh a x$
* Primitive of $\tanh a x$
* Primitive of $\sech a x$
* Primitive of $\csch a x$
== Sources ==
* : $\S 14$: Integrals involving $\coth a x$: $14.615$
* : Back endpapers: A Brief Table of Integrals: $124$.
* : $\S 17$: Tables of Special Indefinite Integrals: $(31)$ Integrals Involving $\coth a x$: $17.31.1.$
Category:Primitives involving Hyperbolic Cotangent Function
Primitive of Hyperbolic Cotangent of a x over x

== Theorem ==
where $B_k$ denotes the $k$th Bernoulli number.
== Proof ==
== Also see ==
* Primitive of $\dfrac {\sinh a x} x$
* Primitive of $\dfrac {\cosh a x} x$
* Primitive of $\dfrac {\tanh a x} x$
* Primitive of $\dfrac {\sech a x} x$
* Primitive of $\dfrac {\csch a x} x$
== Sources ==
* : $\S 14$: Integrals involving $\coth a x$: $14.623$
* : $\S 17$: Tables of Special Indefinite Integrals: $(31)$ Integrals Involving $\coth a x$: $17.31.6.$
Category:Primitives involving Hyperbolic Cotangent Function
Primitive of Hyperbolic Secant Function

== Theorem ==
=== Arcsine Form ===
=== Arctangent of Hyperbolic Sine Form ===
=== Arctangent of Exponential Form ===
=== Arctangent of Half Hyperbolic Tangent Form ===
== Also see ==
* Primitive of Hyperbolic Sine Function
* Primitive of Hyperbolic Cosine Function
* Primitive of Hyperbolic Tangent Function
* Primitive of Hyperbolic Cotangent Function
* Primitive of Hyperbolic Cosecant Function
== Sources ==
* : Chapter $10$: Integration: $10.4$. Standard integrals: Other Standard Results: $\text {(xxvii)}$
* : $\S 14$: General Rules of Integration: $14.29$
* : $\S 16$: Indefinite Integrals: General Rules of Integration: $16.29.$
Category:Primitive of Hyperbolic Secant Function
Category:Primitives of Hyperbolic Functions
Category:Hyperbolic Secant Function
Category:Primitives involving Hyperbolic Secant Function
Primitive of Hyperbolic Secant Function/Arcsine Form

== Theorem ==
:$\ds \int \sech x \rd x = \map \arcsin {\tanh x} + C$
== Proof ==
Let:
Let:
Then:
== Also see ==
* Primitive of Hyperbolic Secant Function/Arctangent of Exponential Form
* Primitive of Hyperbolic Cosecant Function
== Sources ==
* : $\S 14$: General Rules of Integration: $14.29$
* : Appendix $2$: Table of derivatives and integrals of common functions: Hyperbolic functions
* : $\S 16$: Indefinite Integrals: General Rules of Integration: $16.29.$
Category:Primitive of Hyperbolic Secant Function
Primitive of Hyperbolic Secant Function/Arctangent of Exponential Form

== Theorem ==
:$\ds \int \sech x \rd x = 2 \map \arctan {e^x} + C$
== Proof ==
Let:
Let:
Then:
== Also see ==
* Primitive of Hyperbolic Secant Function/Arcsine Form
* Primitive of Hyperbolic Cosecant Function
== Sources ==
* : Chapter $10$: Integration: $10.4$. Standard integrals: Other Standard Results: $\text {(xxvii)}$
* : $\S 14$: General Rules of Integration: $14.29$
* : Appendix: Table $2$: Integrals
* : Appendix: Table $2$: Integrals
* : $\S 16$: Indefinite Integrals: General Rules of Integration: $16.29.$
Category:Primitive of Hyperbolic Secant Function
Primitive of Hyperbolic Secant Function/Arctangent of Half Hyperbolic Tangent Form

== Theorem ==
:$\ds \int \sech x \rd x = 2 \map \arctan {\tanh \dfrac x 2} + C$
== Proof ==
Let $u = \tanh \dfrac x 2$.
Then:
== Sources ==
* : Chapter $10$: Integration: $10.4$. Standard integrals: Other Standard Results: $\text {(xxvii)}$
Category:Primitive of Hyperbolic Secant Function
Category:Hyperbolic Tangent Half-Angle Substitutions
Primitive of Hyperbolic Secant Function/Arctangent of Hyperbolic Sine Form

== Theorem ==
:$\ds \int \sech x \rd x = \map \arctan {\sinh x} + C$
== Proof ==
We have that:
Let:
Then:
== Sources ==
* : Chapter $5$. Integrals: Integrals of Special Functions: $19$
Category:Primitive of Hyperbolic Secant Function
Primitive of Hyperbolic Secant of a x

== Theorem ==
=== Arcsine Form ===
=== Arctangent of Hyperbolic Sine Form ===
=== Arctangent of Exponential Form ===
== Also see ==
* Primitive of $\sinh a x$
* Primitive of $\cosh a x$
* Primitive of $\tanh a x$
* Primitive of $\coth a x$
* Primitive of $\csch a x$
Category:Primitives involving Hyperbolic Secant Function
Primitive of Hyperbolic Secant of a x/Arcsine Form

== Theorem ==
:$\ds \int \sech a x \rd x = \frac {\map \arcsin {\tanh a x} } a + C$
== Proof ==
== Sources ==
* : Back endpapers: A Brief Table of Integrals: $129$.
Category:Primitives involving Hyperbolic Secant Function
Primitive of Hyperbolic Secant of a x/Arctangent of Exponential Form

== Theorem ==
:$\ds \int \sech a x \rd x = \frac {2 \map \arctan {e^{a x} } } a + C$
== Proof ==
== Sources ==
* : $\S 14$: Integrals involving $\sech a x$: $14.626$
* : $\S 17$: Tables of Special Indefinite Integrals: $(33)$ Integrals Involving $\sech a x$: $17.32.1.$
Category:Primitives involving Hyperbolic Secant Function
Primitive of Hyperbolic Secant of a x over x

== Theorem ==
where $E_k$ denotes the $k$th Euler number.
== Proof ==
== Also see ==
* Primitive of $\dfrac {\sinh a x} x$
* Primitive of $\dfrac {\cosh a x} x$
* Primitive of $\dfrac {\tanh a x} x$
* Primitive of $\dfrac {\coth a x} x$
* Primitive of $\dfrac {\csch a x} x$
== Sources ==
* : $\S 14$: Integrals involving $\sech a x$: $14.633$
* : $\S 17$: Tables of Special Indefinite Integrals: $(32)$ Integrals Involving $\sech a x$: $17.32.6.$
Category:Primitives involving Hyperbolic Secant Function
Primitive of Hyperbolic Sine Function

== Theorem ==
:$\ds \int \sinh x \rd x = \cosh x + C$
where $C$ is an arbitrary constant.
== Proof ==
From Derivative of Hyperbolic Cosine:
:$\map {\dfrac \d {\d x} } {\cosh x} = \sinh x$
The result follows from the definition of primitive.
== Also see ==
* Primitive of Hyperbolic Cosine Function
* Primitive of Hyperbolic Tangent Function
* Primitive of Hyperbolic Cotangent Function
* Primitive of Hyperbolic Secant Function
* Primitive of Hyperbolic Cosecant Function
== Sources ==
* : Chapter $\text {II}$: Integration of Elementary Functions: $\S 7$. Standard Integrals: $9$.
* : Chapter $\text {III}$: Integration: Integration
* : Chapter $10$: Integration: $10.4$. Standard integrals: Other Standard Results: $\text {(xxiii)}$
* : $\S 14$: General Rules of Integration: $14.25$
* : Chapter $5$. Integrals: Integrals of Special Functions: $15$
* : $6$. Integral Calculus: Appendix: Table of Fundamental Standard Integrals
* : Appendix $2$: Table of derivatives and integrals of common functions: Hyperbolic functions
* : Appendix: Table $2$: Integrals
* : Appendix: Table $2$: Integrals
* : $\S 16$: Indefinite Integrals: General Rules of Integration: $16.25.$
* : Appendix $7$: Integrals
* : Appendix $8$: Integrals
Category:Primitives of Hyperbolic Functions
Category:Hyperbolic Sine Function
Category:Primitives involving Hyperbolic Sine Function