nyuuzyou/wikis huggingface dataset:
Primitive of Power of x over Power of a squared minus x squared

== Theorem ==
:$\ds \int \frac {x^m \rd x} {\paren {a^2 - x^2}^n} = a^2 \int \frac {x^{m - 2} \rd x} {\paren {a^2 - x^2}^n} - \int \frac {x^{m - 2} \rd x} {\paren {a^2 - x^2}^{n - 1} }$
for $x^2 < a^2$.
== Proof ==
== Also see ==
* Primitive of $\dfrac {x^m} {\paren {x^2 - a^2}^n}$
== Sources ==
* : $\S 14$: Integrals involving $a^2 - x^2$, $x^2 < a^2$: $14.180$
* : $\S 17$: Tables of Special Indefinite Integrals: $(8)$ Integrals Involving $x^2 - a^2$, $x^2 < a^2$: $17.8.18.$
Category:Primitives involving a squared minus x squared
Primitive of Power of x over Power of a x squared plus b x plus c

== Theorem ==
Let $a \in \R_{\ne 0}$.
Then:
== 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:
Meanwhile:
== Sources ==
* : $\S 14$: Integrals involving $a x^2 + bx + c$: $14.275$
Category:Primitives involving a x squared plus b x plus c
Primitive of Power of x over Power of x squared minus a squared

== Theorem ==
:$\ds \int \frac {x^m \rd x} {\paren {x^2 - a^2}^n} = \int \frac {x^{m - 2} \rd x} {\paren {x^2 - a^2}^{n - 1} } + a^2 \int \frac {x^{m - 2} \rd x} {\paren {x^2 - a^2}^n}$
for $x^2 > a^2$.
== Proof ==
== Also see ==
* Primitive of $\dfrac {x^m} {\paren {a^2 - x^2}^n}$
== Sources ==
* : $\S 14$: Integrals involving $x^2 - a^2$, $x^2 > a^2$: $14.161$
* : $\S 17$: Tables of Special Indefinite Integrals: $(7)$ Integrals Involving $x^2 - a^2$, $x^2 > a^2$: $17.7.18.$
Category:Primitives involving x squared minus a squared
Primitive of Power of x over Power of x squared plus a squared

== Theorem ==
:$\ds \int \frac {x^m \rd x} {\paren {x^2 + a^2}^n} = \int \frac {x^{m - 2} \rd x} {\paren {x^2 + a^2}^{n - 1} } - a^2 \int \frac {x^{m - 2} \rd x} {\paren {x^2 + a^2}^n}$
== Proof ==
== Sources ==
* : $\S 14$: Integrals involving $x^2 + a^2$: $14.142$
* : $\S 17$: Tables of Special Indefinite Integrals: $(6)$ Integrals Involving $x^2 + a^2$: $17.6.18.$
Category:Primitives involving x squared plus a squared
Primitive of Power of x over Root of a x + b

== Theorem ==
:$\ds \int \frac {x^m} {\sqrt {a x + b} } \rd x = \frac {2 x^m \sqrt{a x + b} } {\paren {2 m + 1} a} - \frac {2 m b} {\paren {2 m + 1} a} \int \frac {x^{m - 1} } {\sqrt {a x + b} } \rd x$
== Proof ==
From Primitive of Power of $x$ by Power of $a x + b$: Decrement of Power of $x$:
:$\ds \int x^m \paren {a x + b}^n \rd x = \frac {x^m \paren {a x + b}^{n + 1} } {\paren {m + n + 1} a} - \frac {m b} {\paren {m + n + 1} a} \int x^{m - 1} \paren {a x + b}^n \rd x$
Putting $n = -\dfrac 1 2$:
== Sources ==
* : $\S 14$: Integrals involving $\sqrt {a x + b}$: $14.94$
* : $\S 17$: Tables of Special Indefinite Integrals: $(2)$ Integrals Involving $\sqrt {a x + b}$: $17.2.11.$
Category:Primitives involving Root of a x + b
Primitive of Power of x over a x squared plus b x plus c

== Theorem ==
Let $a \in \R_{\ne 0}$.
Then:
:$\ds \int \frac {x^m \rd x} {a x^2 + b x + c} = \frac {x^{m - 1} } {\paren {m - 1} a} - \frac b a \int \frac {x^{m - 1} \rd x} {a x^2 + b x + c} - \frac c a \int \frac {x^{m - 2} \rd x} {a x^2 + b x + c}$
== Proof ==
== Sources ==
* : $\S 14$: Integrals involving $a x^2 + b x + c$: $14.268$
Category:Primitives involving a x squared plus b x plus c
Primitive of Power of x over x cubed plus a cubed

== Theorem ==
:$\ds \int \frac {x^m \rd x} {x^3 + a^3} = \frac {x^{m - 2} } {m - 2} - a^3 \int \frac {x^{m - 3} \rd x} {x^3 + a^3}$ 
== Proof ==
== Sources ==
* : $\S 14$: Integrals involving $x^3 + a^3$: $14.309$
Category:Primitives involving x cubed plus a cubed
Primitive of Product of Cosecant and Cotangent

== Theorem ==
:$\ds \int \csc x \cot x \rd x = -\csc x + C$
where $C$ is an arbitrary constant.
== Proof ==
From Derivative of Cosecant Function:
:$\dfrac \d {\d x} \csc x = -\csc x \cot x$
The result follows from the definition of primitive.
== Sources ==
* : Chapter $\text {III}$: Integration: Integration
* : Chapter $10$: Integration: $10.4$. Standard integrals: Other Standard Results: $\text {(xvi)}$
* : $\S 14$: General Rules of Integration: $14.24$
* : Chapter $25$: Fundamental Integration Formulas: $17$.
* : Chapter $5$. Integrals: Integrals of Special Functions: $12$
* : Appendix: Table $2$: Integrals
* : Appendix: Table $2$: Integrals
* : $\S 16$: Indefinite Integrals: General Rules of Integration: $16.24.$
Category:Primitives involving Cosecant Function
Category:Primitives involving Cotangent Function
Primitive of Product of Hyperbolic Cosecant and Cotangent

== Theorem ==
:$\ds \int \csch x \coth x \rd x = -\csch x + C$
where $C$ is an arbitrary constant.
== Proof ==
From Derivative of Hyperbolic Cosecant:
:$\dfrac \d {\d x} \csch x = -\csch x \coth x$
The result follows from the definition of primitive.
== Sources ==
* : Chapter $10$: Integration: $10.4$. Standard integrals: Other Standard Results: $\text {(xxxii)}$
* : $\S 14$: General Rules of Integration: $14.38$
* : Chapter $5$. Integrals: Integrals of Special Functions: $24$
* : $\S 16$: Indefinite Integrals: General Rules of Integration: $16.38.$
Category:Primitives of Hyperbolic Functions
Category:Hyperbolic Cosecant Function
Category:Hyperbolic Cotangent Function
Primitive of Product of Hyperbolic Secant and Tangent

== Theorem ==
:$\ds \int \sech x \tanh x \rd x = -\sech x + C$
where $C$ is an arbitrary constant.
== Proof ==
From Derivative of Hyperbolic Secant:
:$\dfrac \d {\d x} \sech x = -\sech x \tanh x$
The result follows from the definition of primitive.
== Sources ==
* : Chapter $10$: Integration: $10.4$. Standard integrals: Other Standard Results: $\text {(xxxi)}$
* : $\S 14$: General Rules of Integration: $14.37$
* : Chapter $5$. Integrals: Integrals of Special Functions: $23$
* : $\S 16$: Indefinite Integrals: General Rules of Integration: $16.37.$
Category:Primitives of Hyperbolic Functions
Category:Hyperbolic Secant Function
Category:Hyperbolic Tangent Function
Primitive of Product of Secant and Tangent

== Theorem ==
:$\ds \int \sec x \tan x \rd x = \sec x + C$
where $C$ is an arbitrary constant.
== Proof ==
From Derivative of Secant Function:
:$\dfrac \d {\d x} \sec x = \sec x \tan x$
The result follows from the definition of primitive.
== Sources ==
* : Chapter $\text {III}$: Integration: Integration
* : Chapter $10$: Integration: $10.4$. Standard integrals: Other Standard Results: $\text {(xv)}$
* : Part $\text {III}$: Derivatives and Integrals: Chapter $18$: Integration in Elementary Terms
* : $\S 14$: General Rules of Integration: $14.23$
* : Chapter $25$: Fundamental Integration Formulas: $16$.
* : Chapter $5$. Integrals: Integrals of Special Functions: $11$
* : Appendix: Table $2$: Integrals
* : Appendix: Table $2$: Integrals
* : $\S 16$: Indefinite Integrals: General Rules of Integration: $16.23.$
Category:Primitives involving Tangent Function
Category:Primitives involving Secant Function
Primitive of Reciprocal

== Theorem ==
:$\ds \int \frac {\d x} x = \ln \size x + C$
for $x \ne 0$.
=== Corollary 1 ===
=== Corollary 2 ===
== Proof ==
== Also presented as ==
== Examples ==
== Sources ==
* : Chapter $\text {III}$: Integration: Integration
* : $\text {II}$. Calculus: Integration
* : Chapter $10$: Integration: $10.4$. Standard integrals: Standard Forms: $\text {(ii) (a)}$
* : Appendix $\text I$: Table of Indefinite Integrals $6$.
* : $6$. Integral Calculus: Appendix: Table of Fundamental Standard Integrals
* : Appendix $2$: Table of derivatives and integrals of common functions
* : Appendix: Table $2$: Integrals
* : $\S 5.1$
* : Appendix: Table $2$: Integrals
* : Appendix $7$: Integrals
* : Appendix $8$: Integrals
Category:Primitive of Reciprocal
Category:Primitives involving Reciprocals
Category:Logarithms
Category:Reciprocals
Primitive of Reciprocal/Also presented as

== Primitive of Reciprocal: Also presented as == 
Some sources, when presenting Primitive of Reciprocal, gloss over the case where $x < 0$ and merely present this result as:
:$\ds \int \frac {\d x} x = \ln x + C$
== Sources ==
* : Chapter $\text {III}$: Integration: Integration
* : $\text {II}$. Calculus: Integration
* : Chapter $10$: Integration: $10.4$. Standard integrals: Standard Forms: $\text {(ii) (a)}$
* : Appendix $2$: Table of derivatives and integrals of common functions
* : $\S 5.1$
Category:Primitive of Reciprocal
Primitive of Reciprocal/Corollary

== Corollary to Primitive of Reciprocal ==
:$\ds \int \frac {\d x} x = \ln x + C$
for $x > 0$.
== Proof ==
From Primitive of Reciprocal:
:$\ds \int \frac {\d x} x = \ln \size x + C$
for $x \ne 0$.
By definition of absolute value:
:$\forall x \in \R_{>0}: \size x = x$
Hence the result.
Category:Primitive of Reciprocal
Primitive of Reciprocal/Corollary 2

== Corollary to Primitive of Reciprocal ==
:$\dfrac {\d} {\d x} \ln \size x = \dfrac 1 x$
for $x \ne 0$.
== Proof ==
Follows directly from Primitive of Reciprocal.
== Sources ==
* : $\S 5.1$
Category:Primitive of Reciprocal
Primitive of Reciprocal/Examples/x minus 5 from 2 to 3

== Example of Use of Primitive of Reciprocal ==
:$\ds \int_2^3 \dfrac {\d x} {x - 5} = \ln \dfrac 2 3$
== Proof ==
From Primitive of Reciprocal and using appropriate substitution:
:$\ds \int \dfrac {\d x} {x - 5} = \ln \size {x - 5} + C$
$\dfrac 1 {x - 5}$ between $2$ and $3$ is negative.
So between $2$ and $3$:
:$\ds \int \dfrac {\d x} {x - 5} = \map \ln {5 - x} + C$
Hence:
== Sources ==
* : $\text {II}$. Calculus: Integral of $1 / x$
Category:Examples of Use of Primitive of Reciprocal
Primitive of Reciprocal/Proof

== Theorem ==
== Proof ==
Suppose $x > 0$.
Then:
:$\ln \size x = \ln x$
The result follows from Derivative of Natural Logarithm Function and the definition of primitive.
Suppose $x < 0$.
Then:
and the result again follows from the definition of the primitive.
== Sources ==
* : Chapter $\text {II}$: Integration of Elementary Functions: $\S 7$. Standard Integrals: $2$.
* : $\text {II}$. Calculus: Integration
* : Part $\text {III}$: Derivatives and Integrals: Chapter $18$: Integration in Elementary Terms
* : Chapter $25$: Fundamental Integration Formulas: $5$.
* : Chapter $5$. Integrals: Integrals of Special Functions: $2$
Category:Primitive of Reciprocal
Primitive of Reciprocal of 1 minus Cosine of x

== Theorem ==
:$\ds \int \frac {\d x} {1 - \cos x} = -\cot \frac x 2 + C$
=== Corollary ===
== Proof ==
== Also see ==
* Primitive of $\dfrac 1 {1 + \cos x}$
== Sources ==
* : $6$. Integral Calculus: Appendix: Table of Fundamental Standard Integrals
* : Appendix: Table $2$: Integrals
* : Appendix: Table $2$: Integrals
Category:Primitive of Reciprocal of 1 minus Cosine of x
Category:Primitives involving Cosine Function
Primitive of Reciprocal of 1 minus Cosine of x/Corollary

== Theorem ==
:$\ds \int \frac {\d x} {1 - \cos a x} = -\frac 1 a \cot \frac {a x} 2 + C$
== Proof ==
== Also see ==
* Primitive of $\dfrac 1 {1 + \cos a x}$
* Primitive of $\dfrac 1 {1 + \sin a x}$
* Primitive of $\dfrac 1 {1 - \sin a x}$
== Sources ==
* : $\S 14$: Integrals involving $\cos a x$: $14.384$
* : Back endpapers: A Brief Table of Integrals: $77$.
Category:Primitive of Reciprocal of 1 minus Cosine of x
Primitive of Reciprocal of 1 minus Sine of a x

== Theorem ==
:$\ds \int \frac {\d x} {1 - \sin a x} = \frac 1 a \map \tan {\frac \pi 4 + \frac {a x} 2} + C$
=== Corollary 1 ===
=== Corollary 2 ===
== Proof ==
Then:
== Also see ==
* Primitive of $\dfrac 1 {1 + \sin a x}$
* Primitive of $\dfrac 1 {1 + \cos a x}$
* Primitive of $\dfrac 1 {1 - \cos a x}$
== Sources ==
* : $\S 14$: Integrals involving $\sin a x$: $14.354$
* : Back endpapers: A Brief Table of Integrals: $73$.
Category:Primitive of Reciprocal of 1 minus Sine of a x
Category:Primitives involving Sine Function
Primitive of Reciprocal of 1 minus Sine of a x/Corollary 1

== Theorem ==
:$\ds \int \frac {\d x} {1 - \sin x} = \map \tan {\frac x 2 + \frac \pi 4} + C$
== Proof ==
== Sources ==
* : $6$. Integral Calculus: Appendix: Table of Fundamental Standard Integrals
Category:Primitive of Reciprocal of 1 minus Sine of a x
Primitive of Reciprocal of 1 minus Sine of a x/Corollary 2

== Theorem ==
:$\ds \int \frac {\d x} {1 - \sin x} = -\map \cot {\frac x 2 - \frac \pi 4} + C$
== Proof ==
== Sources ==
* : $6$. Integral Calculus: Appendix: Table of Fundamental Standard Integrals
Category:Primitive of Reciprocal of 1 minus Sine of a x
Primitive of Reciprocal of 1 plus Cosine of x

== Theorem ==
:$\ds \int \frac {\d x} {1 + \cos x} = \tan \frac x 2 + C$
=== Corollary ===
== Proof ==
== Also see ==
* Primitive of $\dfrac 1 {1 - \cos x}$
== Sources ==
* : $6$. Integral Calculus: Appendix: Table of Fundamental Standard Integrals
* : Appendix: Table $2$: Integrals
* : Appendix: Table $2$: Integrals
Category:Primitive of Reciprocal of 1 plus Cosine of x
Category:Primitives involving Cosine Function
Primitive of Reciprocal of 1 plus Cosine of x/Corollary

== Theorem ==
:$\ds \int \frac {\d x} {1 + \cos a x} = \frac 1 a \tan \frac {a x} 2 + C$
== Proof ==
== Also see ==
* Primitive of $\dfrac 1 {1 - \cos a x}$
* Primitive of $\dfrac 1 {1 + \sin a x}$
* Primitive of $\dfrac 1 {1 - \sin a x}$
== Sources ==
* : $\S 14$: Integrals involving $\cos a x$: $14.386$
* : Back endpapers: A Brief Table of Integrals: $76$.
Category:Primitive of Reciprocal of 1 plus Cosine of x
Primitive of Reciprocal of 1 plus Sine of a x

== Theorem ==
:$\ds \int \frac {\d x} {1 + \sin a x} = -\frac 1 a \map \tan {\frac \pi 4 - \frac {a x} 2} + C$
=== Corollary ===
== Proof ==
Then:
== Also see ==
* Primitive of $\dfrac 1 {1 - \sin a x}$
* Primitive of $\dfrac 1 {1 + \cos a x}$
* Primitive of $\dfrac 1 {1 - \cos a x}$
== Sources ==
* : $\S 14$: Integrals involving $\sin a x$: $14.356$
* : Back endpapers: A Brief Table of Integrals: $72$.
Category:Primitive of Reciprocal of 1 plus Sine of a x
Category:Primitives involving Sine Function
Primitive of Reciprocal of 1 plus Sine of a x/Corollary

== Theorem ==
:$\ds \int \frac {\d x} {1 + \sin x} = \map \tan {\frac x 2 - \frac \pi 4} + C$
== Proof ==
== Sources ==
* : $6$. Integral Calculus: Appendix: Table of Fundamental Standard Integrals
Category:Primitive of Reciprocal of 1 plus Sine of a x
Primitive of Reciprocal of 1 plus x squared/Arctangent Form/Proof 1

== Corollary to Primitive of $\frac 1 {x^2 + a^2}$: Arctangent Form ==
== Proof ==
From Primitive of $\dfrac 1 {x^2 + a^2}$: Arctangent Form:
:$\ds \int \frac {\d x} {x^2 + a^2} = \frac 1 a \arctan \frac x a + C$
The result follows by setting $a = 1$.
Category:Primitive of Reciprocal of x squared plus a squared
Primitive of Reciprocal of Cosecant of a x

== Theorem ==
:$\ds \int \frac {\d x} {\csc a x} = \frac {-\cos a x} a + C$
== Proof ==
== Also see ==
* Primitive of $\dfrac 1 {\sin a x}$
* Primitive of $\dfrac 1 {\cos a x}$
* Primitive of $\dfrac 1 {\tan a x}$
* Primitive of $\dfrac 1 {\cot a x}$
* Primitive of $\dfrac 1 {\sec a x}$
== Sources ==
* : $\S 14$: Integrals involving $\csc a x$: $14.465$
Category:Primitives involving Cosecant Function
Primitive of Reciprocal of Cosine of a x

== Theorem ==
=== Logarithm of Secant plus Tangent Form ===
=== Logarithm of Tangent Form ===
== Also see ==
* Primitive of $\dfrac 1 {\sin a x}$
* Primitive of $\dfrac 1 {\tan a x}$
* Primitive of $\dfrac 1 {\cot a x}$
* Primitive of $\dfrac 1 {\sec a x}$
* Primitive of $\dfrac 1 {\csc a x}$
Category:Primitives involving Cosine Function
Primitive of Reciprocal of Cosine of a x/Logarithm of Secant plus Tangent Form

== Theorem ==
:$\ds \int \frac {\d x} {\cos a x} = \frac 1 a \ln \size {\sec a x + \tan a z}$
== Proof ==
== Also see ==
* Primitive of $\dfrac 1 {\sin a x}$: $\csc - \cot$ form
== Sources ==
* : $\S 14$: Integrals involving $\cos a x$: $14.375$
Category:Primitives involving Cosine Function
Primitive of Reciprocal of Cosine of a x/Logarithm of Tangent Form

== Theorem ==
:$\ds \int \frac {\d x} {\cos a x} = \frac 1 a \ln \size {\map \tan {\frac \pi 4 + \frac {a x} 2} } + C$
== Proof ==
== Also see ==
* Primitive of $\dfrac 1 {\sin a x}$: $\tan$ form
== Sources ==
* : $\S 14$: Integrals involving $\cos a x$: $14.375$
Category:Primitives involving Cosine Function
Primitive of Reciprocal of Cosine of a x by 1 minus Sine of a x

== Theorem ==
:$\ds \int \frac {\d x} {\cos a x \paren {1 - \sin a x} } = \frac 1 {2 a \paren {1 - \sin a x} } + \frac 1 {2 a} \ln \size {\map \tan {\frac {a x} 2 + \frac \pi 4} } + C$
== Proof ==
Let:
Then:
== Also see ==
* Primitive of $\dfrac 1 {\cos a x \paren {1 + \sin a x} }$
* Primitive of $\dfrac 1 {\sin a x \paren {1 + \cos a x} }$
* Primitive of $\dfrac 1 {\sin a x \paren {1 - \cos a x} }$
== Sources ==
* : $\S 14$: Integrals involving $\sin a x$ and $\cos a x$: $14.410$
Category:Primitives involving Sine Function and Cosine Function
Primitive of Reciprocal of Cosine of a x by 1 plus Sine of a x

== Theorem ==
:$\ds \int \frac {\d x} {\cos a x \paren {1 + \sin a x} } = \frac {-1} {2 a \paren {1 + \sin a x} } + \frac 1 {2 a} \ln \size {\map \tan {\frac {a x} 2 + \frac \pi 4} } + C$
== Proof ==
Let:
Then:
== Also see ==
* Primitive of $\dfrac 1 {\cos a x \paren {1 - \sin a x} }$
* Primitive of $\dfrac 1 {\sin a x \paren {1 + \cos a x} }$
* Primitive of $\dfrac 1 {\sin a x \paren {1 - \cos a x} }$
== Sources ==
* : $\S 14$: Integrals involving $\sin a x$ and $\cos a x$: $14.410$
Category:Primitives involving Sine Function and Cosine Function
Primitive of Reciprocal of Cosine of a x by 1 plus Sine of a x/Weierstrass Substitution

== Lemma for Primitive of Reciprocal of $\cos a x \paren {1 + \sin a x}$ ==
The Weierstrass Substitution of $\ds \int \frac {\d x} {\cos a x \paren {1 + \sin a x} }$ is:
:$\ds \frac 2 a \int \frac {\d u} {\paren {1 - u} \paren {1 + u}^3}$
where $u = \tan \dfrac {a x} 2$.
== Proof ==
Category:Primitives involving Sine Function and Cosine Function
Category:Weierstrass Substitutions
Primitive of Reciprocal of Cotangent of a x

== Theorem ==
:$\ds \int \frac {\d x} {\cot a x} = \frac {-\ln \size {\cos a x} } a + C$
== Proof ==
== Also see ==
* Primitive of $\dfrac 1 {\sin a x}$
* Primitive of $\dfrac 1 {\cos a x}$
* Primitive of $\dfrac 1 {\tan a x}$
* Primitive of $\dfrac 1 {\sec a x}$
* Primitive of $\dfrac 1 {\csc a x}$
== Sources ==
* : $\S 14$: Integrals involving $\cot a x$: $14.445$
Category:Primitives involving Cotangent Function
Primitive of Reciprocal of Cube of Cosine of a x

== Theorem ==
:$\ds \int \frac {\d x} {\cos^3 a x} = \frac {\sin a x} {2 a \cos^2 a x} + \frac 1 {2 a} \ln \size {\map \tan {\frac \pi 4 + \frac {a x} 2} } + C$
== Proof ==
== Also see ==
* Primitive of $\dfrac 1 {\sin^3 a x}$
== Sources ==
* : $\S 14$: Integrals involving $\cos a x$: $14.382$
Category:Primitives involving Cosine Function
Primitive of Reciprocal of Cube of Root of a x squared plus b x plus c

== Theorem ==
Let $a \in \R_{\ne 0}$.
Then:
:$\ds \int \frac {\d x} {\paren {\sqrt {a x^2 + b x + c} }^3} = \frac {2 \paren {2 a x + b} } {\paren {4 a c - b^2} \sqrt {a x^2 + b x + c} } + C$ 
== Proof ==
For $a > 0$:
Then:
Let $4 a c - b^2 > 0$.
Then:
Let $4 a c - b^2 < 0$.
Then:
Thus in both cases the same result applies, and so:
== Sources ==
* : $\S 14$: Integrals involving $\sqrt {a x^2 + bx + c}$: $14.290$
Category:Primitives involving Root of a x squared plus b x plus c
Primitive of Reciprocal of Cube of Sine of a x

== Theorem ==
:$\ds \int \frac {\d x} {\sin^3 a x} = \frac {-\cos a x} {2 a \sin^2 a x} + \frac 1 {2 a} \ln \size {\tan \frac {a x} 2} + C$
== Proof ==
== Also see ==
* Primitive of $\dfrac 1 {\cos^3 a x}$
== Sources ==
* : $\S 14$: Integrals involving $\sin a x$: $14.352$
Category:Primitives involving Sine Function
Primitive of Reciprocal of Half Integer Power of a x squared plus b x plus c

== Theorem ==
Let $a \in \R_{\ne 0}$.
Then:
:$\ds \int \frac {\d x} {\paren {a x^2 + b x + c}^{n + \frac 1 2} } = \frac {2 \paren {2 a x + b} } {\paren {2 n - 1} \paren {4 a c - b^2} \paren {a x^2 + b x + c}^{n - \frac 1 2} } + \frac {8 a \paren {n - 1} } {\paren {2 n - 1} \paren {4 a c - b^2} } \int \frac {\d x} {\paren {a x^2 + b x + c}^{n - \frac 1 2} }$ 
== Proof ==
Let:
Also let $q = 4 a c - b^2$.
Then:
From Primitive of $\dfrac 1 {\paren {p x + q}^n \sqrt {a x + b} }$:
:$\ds \int \frac {\d x} {\paren {p x + q}^n \sqrt {a x + b} } = \frac {\sqrt {a x + b} } {\paren {n - 1} \paren {a q - b p} \paren {p x + q}^{n - 1} } + \frac {\paren {2 n - 3} a} {2 \paren {n - 1} \paren {a q - b p} } \int \frac {\d x} {\paren {p x + q}^{n - 1} \sqrt {a x + b} }$
Here $p = 1, a = 1, b = 0$ and $n := n + \dfrac 1 2$:
== Sources ==
* : $\S 14$: Integrals involving $\sqrt {a x^2 + b x + c}$: $14.297$
Category:Primitives involving Root of a x squared plus b x plus c
Primitive of Reciprocal of Hyperbolic Cosecant of a x

== Theorem ==
:$\ds \int \frac {\d x} {\csch a x} = \frac {\cosh a x} a + C$
== Proof ==
== Also see ==
* Primitive of $\dfrac 1 {\sinh a x}$
* Primitive of $\dfrac 1 {\cosh a x}$
* Primitive of $\dfrac 1 {\tanh a x}$
* Primitive of $\dfrac 1 {\coth a x}$
* Primitive of $\dfrac 1 {\sech a x}$
== Sources ==
* : $\S 14$: Integrals involving $\csch a x$: $14.640$
Category:Primitives involving Hyperbolic Cosecant Function
Primitive of Reciprocal of Hyperbolic Cosine of a x

== Theorem ==
:$\ds \int \frac {\d x} {\cosh a x} = \frac {2 \map \arctan {e^{a x} } } a + C$
== Proof ==
== Also see ==
* Primitive of $\dfrac 1 {\sinh a x}$
* Primitive of $\dfrac 1 {\tanh a x}$
* Primitive of $\dfrac 1 {\coth a x}$
* Primitive of $\dfrac 1 {\sech a x}$
* Primitive of $\dfrac 1 {\csch a x}$
== Sources ==
* : $\S 14$: Integrals involving $\cosh a x$: $14.567$
Category:Primitives involving Hyperbolic Cosine Function
Primitive of Reciprocal of Hyperbolic Cosine of a x by 1 plus Hyperbolic Sine of a x

== Theorem ==
:$\ds \int \frac {\rd x} {\cosh a x \paren {1 + \sinh a x} } = \frac 1 {2 a} \ln \size {\frac {1 + \sinh a x} {\cosh a x} } + \frac 1 a \map \arctan {e^{a x} } + C$
== Proof ==
== Sources ==
* : $\S 14$: Integrals involving $\sinh a x$ and $\cosh a x$: $14.601$
Category:Primitives involving Hyperbolic Sine Function
Category:Primitives involving Hyperbolic Cosine Function
Primitive of Reciprocal of Hyperbolic Cosine of a x minus 1

== Theorem ==
:$\ds \int \frac {\d x} {\cosh a x - 1} = \frac {-1} a \coth \frac {a x} 2 + C$
== Proof ==
== Also see ==
* Primitive of $\dfrac 1 {\cosh a x + 1}$
== Sources ==
* : $\S 14$: Integrals involving $\cosh a x$: $14.576$
Category:Primitives involving Hyperbolic Cosine Function
Category:Hyperbolic Tangent Half-Angle Substitutions
Primitive of Reciprocal of Hyperbolic Cosine of a x plus 1

== Theorem ==
:$\ds \int \frac {\d x} {\cosh a x + 1} = \frac 1 a \tanh \frac {a x} 2 + C$
== Proof ==
== Also see ==
* Primitive of $\dfrac 1 {\cosh a x - 1}$
== Sources ==
* : $\S 14$: Integrals involving $\cosh a x$: $14.575$
Category:Primitives involving Hyperbolic Cosine Function
Category:Hyperbolic Tangent Half-Angle Substitutions
Primitive of Reciprocal of Hyperbolic Cotangent of a x

== Theorem ==
:$\ds \int \frac {\d x} {\coth a x} = \frac {\ln \size {\cosh a x} } a + C$
== Proof ==
== Also see ==
* Primitive of $\dfrac 1 {\sinh a x}$
* Primitive of $\dfrac 1 {\cosh a x}$
* Primitive of $\dfrac 1 {\tanh a x}$
* Primitive of $\dfrac 1 {\sech a x}$
* Primitive of $\dfrac 1 {\csch a x}$
== Sources ==
* : $\S 14$: Integrals involving $\coth a x$: $14.620$
Category:Primitives involving Hyperbolic Cotangent Function
Primitive of Reciprocal of Hyperbolic Secant of a x

== Theorem ==
:$\ds \int \frac {\d x} {\sech a x} = \frac {\sinh a x} a + C$
== Proof ==
== Also see ==
* Primitive of $\dfrac 1 {\sinh a x}$
* Primitive of $\dfrac 1 {\cosh a x}$
* Primitive of $\dfrac 1 {\tanh a x}$
* Primitive of $\dfrac 1 {\coth a x}$
* Primitive of $\dfrac 1 {\csch a x}$
== Sources ==
* : $\S 14$: Integrals involving $\sech a x$: $14.630$
Category:Primitives involving Hyperbolic Secant Function
Primitive of Reciprocal of Hyperbolic Sine of a x

== Theorem ==
:$\ds \int \frac {\d x} {\sinh a x} = \frac 1 a \ln \size {\tanh \frac {a x} 2} + C$
== Proof ==
== Also see ==
* Primitive of $\dfrac 1 {\cosh a x}$
* Primitive of $\dfrac 1 {\tanh a x}$
* Primitive of $\dfrac 1 {\coth a x}$
* Primitive of $\dfrac 1 {\sech a x}$
* Primitive of $\dfrac 1 {\csch a x}$
== Sources ==
* : $\S 14$: Integrals involving $\sinh a x$: $14.545$
Category:Primitives involving Hyperbolic Sine Function