nyuuzyou/wikis huggingface dataset:
Primitive of Cosecant Function/Cosecant minus Cotangent Form/Mistake

== Source Work ==
:$\text {II}$. Calculus: Integration
:Chapter $10$: Integration: $10.4$. Standard integrals: Other Standard Results: $\text {(xix)}$
:Appendix $2$: Table of derivatives and integrals of common functions: Trigonometric functions
:Table 2 Integrals
== Mistake ==
:$\ds \int \csc x \rd x = \map \ln {\csc x - \cot x} + C$
== Correction ==
Should be:
:$\ds \int \csc x \rd x = \ln \size {\csc x - \cot x} + C$
It also needs to specify that the result is valid only where $\csc x - \cot x \ne 0$.
== Sources ==
* : $\text {II}$. Calculus: Integration
* : Chapter $10$: Integration: $10.4$. Standard integrals: Other Standard Results: $\text {(xx)}$
* : Appendix $2$: Table of derivatives and integrals of common functions: Trigonometric functions
* : Appendix: Table $2$: Integrals
* : Appendix: Table $2$: Integrals
Category:Mistakes/A Note Book in Pure Mathematics
Category:Mistakes/A Course in Pure Mathematics
Category:Mistakes/Dictionary of Mathematics
Category:Mistakes/The Penguin Dictionary of Mathematics
Primitive of Cosecant Function/Cosecant plus Cotangent Form

== Theorem ==
:$\ds \int \csc x \rd x = -\ln \size {\csc x + \cot x} + C$
where $\csc x + \cot x \ne 0$.
== Proof ==
== Sources ==
* : Chapter $\text {III}$: Integration: Integration
* : $\S 5.2$
Category:Primitive of Cosecant Function
Primitive of Cosecant Function/Cosecant plus Cotangent Form/Mistake

== Source Work ==
:Chapter $\text {III}$: Integration: Integration
== Mistake ==
:$\ds \int \csc x \rd x = -\map \ln {\csc x + \cot x} + C$
== Correction ==
Should be:
:$\ds \int \csc x \rd x = -\ln \size {\csc x + \cot x} + C$
It also needs to specify that the result is valid only where $\csc x + \cot x \ne 0$.
== Sources ==
* : Chapter $\text {III}$: Integration: Integration
Category:Mistakes/Advanced Mathematics for Technical Students, Part I
Primitive of Cosecant Function/Tangent Form

== Theorem ==
:$\ds \int \csc x \rd x = \ln \size {\tan \frac x 2} + C$
where $\tan \dfrac x 2 \ne 0$.
== Proof 1 ==
== Proof 2 ==
== Sources ==
* : Chapter $\text {III}$: Integration: Integration
* : $\text {II}$. Calculus: Integration
* : Chapter $10$: Integration: $10.4$. Standard integrals: Other Standard Results: $\text {(xix)}$
* : $\S 14$: General Rules of Integration: $14.16$
* : Chapter $5$. Integrals: Integrals of Special Functions: $8$
* : $6$. Integral Calculus: Appendix: Table of Fundamental Standard Integrals
* : $\S 16$: Indefinite Integrals: General Rules of Integration: $16.16.$
* : Appendix $7$: Integrals
* : Appendix $8$: Integrals
Category:Primitive of Cosecant Function
Primitive of Cosecant Function/Tangent Form/Proof 1

== Theorem ==
== Proof ==
Since:
:$\tan \dfrac x 2 \ne 0$
It follows that $\forall n \in \Z$: 
:$ 2\pi n < x < 2\pi n + \pi$
So:
:$\csc x + \cot x \ne 0$
And:
:$\dfrac {1 + \cos x} {\sin x} \ne 0$ 
Hence:
Category:Primitive of Cosecant Function
Primitive of Cosecant of a x

== Theorem ==
=== Tangent Form ===
=== Cosecant minus Cotangent Form ===
=== Cosecant plus Cotangent Form ===
== Also see ==
* Primitive of $\sin a x$
* Primitive of $\cos a x$
* Primitive of $\tan a x$
* Primitive of $\cot a x$
* Primitive of $\sec a x$
Category:Primitive of Cosecant of a x
Category:Primitives involving Cosecant Function
Primitive of Cosecant of a x/Cosecant minus Cotangent Form

== Theorem ==
:$\ds \int \csc a x \rd x = \frac 1 a \ln \size {\csc a x - \cot a x} + C$
where $\csc a x - \cot a x \ne 0$.
== Proof ==
== Also see ==
* Primitive of $\sec a x$: Secant plus Tangent Form
== Sources ==
* : $\S 14$: Integrals involving $\csc a x$: $14.461$
* : $\S 17$: Tables of Special Indefinite Integrals: $(23)$ Integrals Involving $\csc a x$: $17.23.1.$
Category:Primitive of Cosecant of a x
Primitive of Cosecant of a x/Cosecant plus Cotangent Form

== Theorem ==
:$\ds \int \csc a x \rd x = -\frac 1 a \ln \size {\csc a x + \cot a x} + C$
where $\csc a x + \cot a x \ne 0$.
== Proof ==
== Also see ==
* Primitive of $\sec a x$: Secant plus Tangent Form
== Sources ==
* : Back endpapers: A Brief Table of Integrals: $89$.
Category:Primitive of Cosecant of a x
Primitive of Cosecant of a x/Tangent Form

== Theorem ==
:$\ds \int \csc a x \rd x = \frac 1 a \ln \size {\tan \frac {a x} 2} + C$
where $\tan \dfrac {a x} 2 \ne 0$.
== Proof ==
== Also see ==
* Primitive of $\sec a x$: Tangent Form
== Sources ==
* : $\S 14$: Integrals involving $\csc a x$: $14.461$
* : $\S 17$: Tables of Special Indefinite Integrals: $(23)$ Integrals Involving $\csc a x$: $17.23.1.$
Category:Primitive of Cosecant of a x
Primitive of Cosecant of a x over x

== Theorem ==
:$\ds \int \frac {\csc a x} x \rd x = \frac {-1} {a x} + \frac {a x} 6 + \frac {7 \paren {a x}^3} {1080} + \cdots + \frac {\paren {-1}^{n - 1} 2 \paren {2^{2 n - 1} - 1} B_{2 n} \paren {a x}^{2 n - 1} } {\paren {2 n - 1} \paren {2 n}!} + \cdots + C$
where $B_n$ is the $n$th Bernoulli number.
== Proof ==
== Also see ==
* Primitive of $\dfrac {\sin a x} x$
* Primitive of $\dfrac {\cos a x} x$
* Primitive of $\dfrac {\tan a x} x$
* Primitive of $\dfrac {\cot a x} x$
* Primitive of $\dfrac {\sec a x} x$
== Sources ==
* : $\S 14$: Integrals involving $\csc a x$: $14.467$
Category:Primitives involving Cosecant Function
Primitive of Cosine Function

== Theorem ==
:$\ds \int \cos x \rd x = \sin x + C$
where $C$ is an arbitrary constant.
=== Corollary ===
=== Corollary 2 ===
== Proof 1 ==
== Proof 2 ==
== Also see ==
* Primitive of Sine Function
* Primitive of Tangent Function
* Primitive of Cotangent Function
* Primitive of Secant Function
* Primitive of Cosecant Function
== Sources ==
* : Chapter $\text {III}$: Integration: Integration
* : $\text {II}$. Calculus: Integration
* : Chapter $10$: Integration: $10.4$. Standard integrals: Standard Forms: $\text {(v)}$
* : $\S 14$: General Rules of Integration: $14.12$
* : Front endpapers: A Brief Table of Integrals: $3$.
* : $6$. Integral Calculus: Appendix: Table of Fundamental Standard Integrals
* 
* : Appendix $2$: Table of derivatives and integrals of common functions: Trigonometric functions
* : Appendix: Table $2$: Integrals
* : Appendix: Table $2$: Integrals
* : $\S 16$: Indefinite Integrals: General Rules of Integration: $16.12.$
* : Appendix $7$: Integrals
* : Appendix $8$: Integrals
Category:Primitive of Cosine Function
Category:Primitives of Trigonometric Functions
Category:Primitives involving Cosine Function
Category:Cosine Function
Primitive of Cosine Function/Corollary

== Corollary to Primitive of Cosine Function ==
:$\ds \int \cos a x \rd x = \frac {\sin a x} a + C$
where $a$ is a non-zero constant.
== Proof ==
== Also see ==
* Primitive of $\sin a x$
* Primitive of $\tan a x$
* Primitive of $\cot a x$
* Primitive of $\sec a x$
* Primitive of $\csc a x$
== Sources ==
* : $\S 14$: Integrals involving $\cos a x$: $14.369$
* : Front endpapers: A Brief Table of Integrals: $57$.
* : $9$ Sums and Integrals: $9.8$ Standard Integrals
* : $\S 17$: Tables of Special Indefinite Integrals: $(18)$ Integrals Involving $\cos a x$: $17.18.1.$
Category:Primitives involving Cosine Function
Primitive of Cosine Function/Proof 1

== Theorem ==
== Proof ==
From Derivative of Sine Function:
:$\dfrac \d {\d x} \sin x = \cos x$
The result follows from the definition of primitive.
== Sources ==
* : Chapter $\text {II}$: Integration of Elementary Functions: $\S 7$. Standard Integrals: $4$.
* : $\text {II}$. Calculus: Integration
* : Part $\text {III}$: Derivatives and Integrals: Chapter $18$: Integration in Elementary Terms
* : Chapter $25$: Fundamental Integration Formulas: $9$.
* : Chapter $5$. Integrals: Integrals of Special Functions: $4$
Category:Primitive of Cosine Function
Primitive of Cosine Integral Function

== Theorem ==
:$\ds \int \map \Ci x \rd x = x \map \Ci x + \sin x + C$
where:
:$\Ci$ denotes the cosine integral function
:$x$ is a strictly positive real number.
== Proof ==
By Derivative of Cosine Integral Function, we have: 
:$\ds \frac \d {\d x} \paren {\map \Ci x} = -\frac {\cos x} x$
So:
Category:Primitives
Category:Cosine Integral Function
Primitive of Cosine of a x + b

== Corollary to Primitive of Cosine Function ==
:$\ds \int \map \cos {a x + b} \rd x = \frac {\map \sin {a x + b} } a + C$
where $a$ is a non-zero constant.
== Proof 1 ==
== Proof 2 ==
== Proof 3 ==
Category:Primitives involving Cosine Function
Category:Primitives involving a x + b
Category:Primitive of Cosine of a x + b
Primitive of Cosine of a x + b/Proof 2

== Corollary to Primitive of Cosine Function ==
== Proof ==
Let $u = a x + b$.
Then:
:$\dfrac {\d u} {\d x} = a$
Then:
== Sources ==
* : Chapter $\text {II}$: Integration of Elementary Functions: $\S 8$. Change of Variable
Category:Primitive of Cosine of a x + b
Primitive of Cosine of a x by Cosine of b x

== Theorem ==
:$\ds \int \cos a x \cos b x \rd x = \frac {\map \sin {\paren {a - b} x} } {2 \paren {a - b} } + \frac {\map \sin {\paren {a + b} x} } {2 \paren {a + b} } + C$
== Proof ==
== Also see ==
* Primitive of $\sin a x \sin b x$
* Primitive of $\sin a x \cos b x$
== Sources ==
* : $\S 14$: Integrals involving $\cos a x$: $14.383$
* : Front endpapers: A Brief Table of Integrals: $62. \ \text {(c)}$
Category:Primitives involving Cosine Function
Primitive of Cosine of a x over Power of p plus q of Sine of a x

== Theorem ==
:$\ds \int \frac {\cos a x \rd x} {\paren {p + q \sin a x}^n} = \frac {-1} {a q \paren {n - 1} \paren {p + q \sin a x}^{n - 1} } + C$
== Proof ==
== Also see ==
* Primitive of $\dfrac {\sin a x} {\paren {p + q \cos a x}^n}$
== Sources ==
* : $\S 14$: Integrals involving $\sin a x$ and $\cos a x$: $14.418$
Category:Primitives involving Sine Function and Cosine Function
Primitive of Cosine of a x over Power of x

== Theorem ==
:$\ds \int \frac {\cos a x} {x^n} \rd x = \frac {-\cos a x} {\paren {n - 1} x^{n - 1} } - \frac a {n - 1} \int \frac {\sin a x} {x^{n - 1} } \rd x$
== 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 $\dfrac {\sin a x} {x^n}$
== Sources ==
* : $\S 14$: Integrals involving $\cos a x$: $14.395$
Category:Primitives involving Cosine Function
Primitive of Cosine of a x over Sine of a x

== Theorem ==
:$\ds \int \dfrac {\cos a x} {\sin a x} \rd x = \dfrac 1 a \ln \size {\sin a x} + C$
== Proof ==
== Also see ==
* Primitive of $\dfrac {\sin a x} {\cos a x}$
== Sources ==
* : Front endpapers: A Brief Table of Integrals: $65$.
Category:Primitives involving Sine Function and Cosine Function
Primitive of Cosine of a x over Sine of a x minus Cosine of a x

== Theorem ==
:$\ds \int \frac {\cos a x \rd x} {\sin a x - \cos a x} = \frac {-x} 2 + \frac 1 {2 a} \ln \size {\sin a x - \cos a x} + C$
== Proof ==
== Also see ==
* Primitive of $\dfrac {\cos a x} {\sin a x + \cos a x}$
* Primitive of $\dfrac {\sin a x} {\sin a x + \cos a x}$
* Primitive of $\dfrac {\sin a x} {\sin a x - \cos a x}$
== Sources ==
* : $\S 14$: Integrals involving $\sin a x$ and $\cos a x$: $14.414$
Category:Primitives involving Sine Function and Cosine Function
Primitive of Cosine of a x over Sine of a x plus Cosine of a x

== Theorem ==
:$\ds \int \frac {\cos a x \rd x} {\sin a x + \cos a x} = \frac x 2 + \frac 1 {2 a} \ln \size {\sin a x + \cos a x} + C$
== Proof ==
== Also see ==
* Primitive of $\dfrac {\cos a x} {\sin a x - \cos a x}$
* Primitive of $\dfrac {\sin a x} {\sin a x + \cos a x}$
* Primitive of $\dfrac {\sin a x} {\sin a x - \cos a x}$
== Sources ==
* : $\S 14$: Integrals involving $\sin a x$ and $\cos a x$: $14.414$
Category:Primitives involving Sine Function and Cosine Function
Primitive of Cosine of a x over Sine of a x plus phi

== Theorem ==
:$\ds \int \frac {\cos a x \rd x} {\map \sin {a x + \phi} } = \frac {\ln \size {\map \sin {a x + \phi} } } {a \cos \phi} + \tan \phi \int \frac {\sin a x \rd x} {\map \sin {a x + \phi} } + C$
== Proof ==
First note that:
Then:
Category:Primitives involving Sine Function
Category:Primitives involving Cosine Function
Primitive of Cosine of a x over p plus q of Sine of a x

== Theorem ==
:$\ds \int \frac {\cos a x \rd x} {p + q \sin a x} = \frac 1 {a q} \ln \size {p + q \sin a x} + C$
== Proof ==
== Also see ==
* Primitive of $\dfrac {\sin a x} {p + q \cos a x}$
== Sources ==
* : $\S 14$: Integrals involving $\sin a x$ and $\cos a x$: $14.416$
Category:Primitives involving Sine Function and Cosine Function
Primitive of Cosine of a x over x

== Theorem ==
== Proof ==
The validity of $(1)$ follows from Cosine Function is Absolutely Convergent.
== Also see ==
* Primitive of $\dfrac {\sin a x} x$
* Primitive of $\dfrac {\tan a x} x$
* Primitive of $\dfrac {\cot a x} x$
* Primitive of $\dfrac {\sec a x} x$
* Primitive of $\dfrac {\csc a x} x$
== Sources ==
* : $\S 14$: Integrals involving $\cos a x$: $14.373$
Category:Primitives involving Cosine Function
Primitive of Cosine of a x over x squared

== Theorem ==
:$\ds \int \frac {\cos a x \rd x} {x^2} = \frac {-\cos a x} x - a \int \frac {\sin a x \rd x} x$
== 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 $\dfrac {\sin a x} x$
* Primitive of $\dfrac {\sin a x} {x^2}$
== Sources ==
* : $\S 14$: Integrals involving $\cos a x$: $14.374$
Category:Primitives involving Cosine Function
Primitive of Cotangent Function

== Theorem ==
:$\ds \int \cot x \rd x = \ln \size {\sin x} + C$
where $\sin x \ne 0$.
== Proof ==
== Also see ==
* Primitive of Sine Function
* Primitive of Cosine Function
* Primitive of Tangent Function
* Primitive of Secant Function
* Primitive of Cosecant Function
== Sources ==
* : Chapter $\text {III}$: Integration: Integration
* : $\text {II}$. Calculus: Integration
* : Chapter $10$: Integration: $10.4$. Standard integrals: Other Standard Results: $\text {(xviii)}$
* : $\S 14$: General Rules of Integration: $14.14$
* : Chapter $25$: Fundamental Integration Formulas: $11$.
* : Chapter $5$. Integrals: Integrals of Special Functions: $6$
* : $6$. Integral Calculus: Appendix: Table of Fundamental Standard Integrals
* : Appendix $2$: Table of derivatives and integrals of common functions: Trigonometric functions
* : Appendix: Table $2$: Integrals
* : Appendix: Table $2$: Integrals
* : $\S 16$: Indefinite Integrals: General Rules of Integration: $16.14.$
* : Appendix $7$: Integrals
* : Appendix $8$: Integrals
Category:Primitive of Cotangent Function
Category:Primitives of Trigonometric Functions
Category:Primitives involving Cotangent Function
Category:Cotangent Function
Primitive of Cotangent of a x

== Theorem ==
:$\ds \int \cot a x \rd x = \frac {\ln \size {\sin a x} } a + C$
== Proof ==
== Examples ==
== Also see ==
* Primitive of $\sin a x$
* Primitive of $\cos a x$
* Primitive of $\tan a x$
* Primitive of $\sec a x$
* Primitive of $\csc a x$
== Sources ==
* : $\S 14$: Integrals involving $\cot a x$: $14.440$
* : Back endpapers: A Brief Table of Integrals: $83$.
* : $\S 17$: Tables of Special Indefinite Integrals: $(21)$ Integrals Involving $\cot a x$: $17.21.1.$
Category:Primitive of Cotangent of a x
Category:Primitive of Cotangent Function
Category:Primitives involving Cotangent Function
Primitive of Cotangent of a x/Examples/Half x

== Example of Use of Primitive of $\cot a x$ ==
:$\ds \int \cot \dfrac x 2 \rd x = 2 \ln \size {\sin \dfrac 1 2} + C$
== Proof ==
From Primitive of $\cot a x$:
Then:
The result follows by setting $a = \dfrac 1 2$.
== Proof ==
* : $\text {II}$. Calculus: Exercises $\text {XIV}$: $3$.
Category:Examples of Use of Primitive of Cotangent of a x
Primitive of Cotangent of a x over x

== Theorem ==
:$\ds \int \frac {\cot a x} x \rd x = \frac {-1} a x - \frac {a x} 3 - \frac {\paren {a x}^3} {135} - \cdots - \frac {\paren {-1}^{n - 1} 2^{2 n} B_{2 n} \paren {a x}^{2 n - 1} } {\paren {2 n - 1} \paren {2 n}!} - \cdots + C$
where $B_n$ denotes the $n$th Bernoulli number.
== Proof ==
== Also see ==
* Primitive of $\dfrac {\sin a x} x$
* Primitive of $\dfrac {\cos a x} x$
* Primitive of $\dfrac {\tan a x} x$
* Primitive of $\dfrac {\sec a x} x$
* Primitive of $\dfrac {\csc a x} x$
== Sources ==
* : $\S 14$: Integrals involving $\cot a x$: $14.447$
Category:Primitives involving Cotangent Function
Primitive of Cube of Cosecant of a x

== Theorem ==
:$\ds \int \csc^3 a x \rd x = \frac {-\csc a x \cot a x} {2 a} + \frac 1 {2 a} \ln \size {\tan \dfrac {a x} 2} + C$
== Proof ==
== Also see ==
* Primitive of $\sin^3 a x$
* Primitive of $\cos^3 a x$
* Primitive of $\tan^3 a x$
* Primitive of $\cot^3 a x$
* Primitive of $\sec^3 a x$
== Sources ==
* : $\S 14$: Integrals involving $\csc a x$: $14.463$
Category:Primitives involving Cosecant Function
Primitive of Cube of Cosine of a x

== Theorem ==
:$\ds \int \cos^3 a x \rd x = \frac {\sin a x} a - \frac {\sin^3 a x} {3 a} + C$
== Proof ==
== Also see ==
* Primitive of $\sin^3 a x$
* Primitive of $\tan^3 a x$
* Primitive of $\cot^3 a x$
* Primitive of $\sec^3 a x$
* Primitive of $\csc^3 a x$
== Sources ==
* : $\S 14$: Integrals involving $\cos a x$: $14.379$
Category:Primitives involving Cosine Function
Primitive of Cube of Cotangent of a x

== Theorem ==
:$\ds \int \cot^3 a x \rd x = \frac {-\cot^2 a x} {2 a} - \frac 1 a \ln \size {\sin a x} + C$
== Proof ==
== Also see ==
* Primitive of $\sin^3 a x$
* Primitive of $\cos^3 a x$
* Primitive of $\tan^3 a x$
* Primitive of $\sec^3 a x$
* Primitive of $\csc^3 a x$
== Sources ==
* : $\S 14$: Integrals involving $\cot a x$: $14.442$
Category:Primitives involving Cotangent Function
Primitive of Cube of Hyperbolic Cosecant of a x

== Theorem ==
:$\ds \int \csch^3 a x \rd x = \frac {-\csch a x \coth a x} {2 a} - \frac 1 {2 a} \ln \size {\tanh a x} + C$
== Proof ==
== Also see ==
* Primitive of $\tanh^3 a x$
* Primitive of $\coth^3 a x$
* Primitive of $\sech^3 a x$
== Sources ==
* : $\S 14$: Integrals involving $\csch a x$: $14.638$
* : $\S 17$: Tables of Special Indefinite Integrals: $(33)$ Integrals Involving $\csch a x$: $17.33.3.$
Category:Primitives involving Hyperbolic Cosecant Function
Primitive of Cube of Hyperbolic Cotangent of a x

== Theorem ==
:$\ds \int \coth^3 a x \rd x = \frac {\ln \size {\sinh a x} } a - \frac {\coth^2 a x} {2 a} + C$
== Proof 1 ==
== Proof 2 ==
== Also see ==
* Primitive of $\tanh^3 a x$
* Primitive of $\sech^3 a x$
* Primitive of $\csch^3 a x$
== Sources ==
* : $\S 14$: Integrals involving $\coth a x$: $14.617$
* : $\S 17$: Tables of Special Indefinite Integrals: $(31)$ Integrals Involving $\coth a x$: $17.31.3.$
Category:Primitive of Cube of Hyperbolic Cotangent of a x
Category:Primitives involving Hyperbolic Cotangent Function
Primitive of Cube of Hyperbolic Secant of a x

== Theorem ==
:$\ds \int \sech^3 a x \rd x = \frac {\sech a x \tanh a x} {2 a} + \frac 1 {2 a} \map \arctan {\sinh a x} + C$
== Proof ==
== Also see ==
* Primitive of $\tanh^3 a x$
* Primitive of $\coth^3 a x$
* Primitive of $\csch^3 a x$
== Sources ==
* : $\S 14$: Integrals involving $\sech a x$: $14.628$
* : $\S 17$: Tables of Special Indefinite Integrals: $(32)$ Integrals Involving $\sech a x$: $17.32.3.$
Category:Primitives involving Hyperbolic Secant Function
Primitive of Cube of Hyperbolic Tangent of a x

== Theorem ==
:$\ds \int \tanh^3 a x \rd x = \frac {\ln \size {\cosh a x} } a - \frac {\tanh^2 a x} {2 a} + C$
== Proof 1 ==
== Proof 2 ==
== Also see ==
* Primitive of $\coth^3 a x$
* Primitive of $\sech^3 a x$
* Primitive of $\csch^3 a x$
== Sources ==
* : $\S 14$: Integrals involving $\tanh a x$: $14.606$
* : $\S 17$: Tables of Special Indefinite Integrals: $(30)$ Integrals Involving $\tanh a x$: $17.30.3.$
Category:Primitive of Cube of Hyperbolic Tangent of a x
Category:Primitives involving Hyperbolic Tangent Function
Primitive of Cube of Secant Function

== Theorem ==
:$\ds \int \sec^3 x \rd x = \frac 1 2 \paren {\sec x \tan x + \ln \size {\sec x + \tan x} } + C$
== Proof 1 ==
== Proof 2 ==
Category:Primitive of Cube of Secant Function
Category:Primitives involving Secant Function
Primitive of Cube of Secant Function/Proof 2

== Theorem ==
== 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:
== Sources ==
* : $\text {II}$. Calculus: Integration: Integration by Parts: Example
Category:Primitive of Cube of Secant Function
Primitive of Cube of Secant of a x

== Theorem ==
:$\ds \int \sec^3 a x \rd x = \frac 1 {2 a} \paren {\sec a x \tan a x + \ln \size {\sec a x + \tan a x} } + C$
== Proof 1 ==
== Proof 2 ==
== Also see ==
* Primitive of $\sin^3 a x$
* Primitive of $\cos^3 a x$
* Primitive of $\tan^3 a x$
* Primitive of $\cot^3 a x$
* Primitive of $\csc^3 a x$
== Sources ==
* : Part $\text {III}$: Derivatives and Integrals: Chapter $18$: Integration in Elementary Terms: Problems $3 \ \text {viii}$
::This is a tricky and important integral that often comes up.
* : $\S 14$: Integrals involving $\sec a x$: $14.453$
Category:Primitive of Cube of Secant Function
Category:Primitives involving Secant Function
Primitive of Cube of Secant of a x/Proof 2

== Theorem ==
== 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:
The result follows.
Category:Primitive of Cube of Secant Function
Primitive of Cube of Sine Function

== Theorem ==
:$\ds \int \sin^3 x \rd x = \frac {\cos^3 x} 3 - \cos x + C$
== Proof 1 ==
== Proof 2 ==
== Sources ==
* : $\text {II}$. Calculus: Exercises $\text {XIV}$: $9$.
Category:Primitive of Cube of Sine Function
Category:Primitives involving Sine Function
Primitive of Cube of Sine of a x

== Theorem ==
:$\ds \int \sin^3 a x \rd x = -\frac {\cos a x} a + \frac {\cos^3 a x} {3 a} + C$
== Proof 1 ==
== Proof 2 ==
== Also see ==
* Primitive of $\cos^3 a x$
* Primitive of $\tan^3 a x$
* Primitive of $\cot^3 a x$
* Primitive of $\sec^3 a x$
* Primitive of $\csc^3 a x$
== Sources ==
* : $\S 14$: Integrals involving $\sin a x$: $14.349$
Category:Primitive of Cube of Sine of a x
Category:Primitives involving Sine Function
Primitive of Cube of Tangent of a x

== Theorem ==
:$\ds \int \tan^3 a x \rd x = \frac {\tan^2 a x} {2 a} + \frac 1 a \ln \size {\cos a x} + C$
== Proof 1 ==
== Proof 2 ==
== Also see ==
* Primitive of $\sin^3 a x$
* Primitive of $\cos^3 a x$
* Primitive of $\cot^3 a x$
* Primitive of $\sec^3 a x$
* Primitive of $\csc^3 a x$
== Sources ==
* : $\S 14$: Integrals involving $\tan a x$: $14.431$
Category:Primitive of Cube of Tangent of a x
Category:Primitives involving Tangent Function
Primitive of Cube of Tangent of x

== Theorem ==
== Proof ==
From Primitive of $\tan^3 a x$:
The result follows on setting $a = 1$, then using Primitive of $\tan x$: Secant Form
== Sources ==
* : $\text {II}$. Calculus: Exercises $\text {XIV}$: $30$.
Category:Primitives involving Tangent Function
Primitive of Error Function

== Theorem ==
:$\ds \int \map \erf x \rd x = x \map \erf x + \frac 1 {\sqrt \pi} e^{-x^2} + C$
where $\erf$ denotes the error function.
== Proof ==
By Derivative of Error Function, we have:
:$\dfrac \d {\d x} \paren {\map \erf x} = \dfrac 2 {\sqrt \pi} e^{-x^2}$
So:
Category:Primitives
Category:Error Function
Primitive of Exponential Function

== Theorem ==
:$\ds \int e^x \rd x = e^x + C$
where $C$ is an arbitrary constant.
=== General Result ===
== Proof for Real Numbers ==
== Proof for Complex Numbers ==
== Examples ==
== Also see ==
* Primitive of $e^{a x}$
== Sources ==
* : Chapter $\text {III}$: Integration: Integration
* : $\text {II}$. Calculus: Integration
* : Part $\text {III}$: Derivatives and Integrals: Chapter $18$: Integration in Elementary Terms
* : $\S 14$: General Rules of Integration: $14.9$
* : Chapter $25$: Fundamental Integration Formulas: $7$.
* : Chapter $5$. Integrals: Integrals of Special Functions: $14$
* : $6$. Integral Calculus: Appendix: Table of Fundamental Standard Integrals
* : Appendix $2$: Table of derivatives and integrals of common functions
* : Appendix: Table $2$: Integrals
* : Appendix: Table $2$: Integrals
* : $\S 16$: Indefinite Integrals: General Rules of Integration: $16.9.$
* 
Category:Primitive of Exponential Function
Category:Primitives involving Exponential Function
Category:Exponential Function