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

== Theorem ==
:$\ds \int \dfrac 1 {x^2} \arccsc \frac x a \rd x = \begin{cases}
-\dfrac 1 x \arccsc \dfrac x a - \dfrac {\sqrt{x^2 - a^2} } {a x} + C & : 0 < \arccsc \dfrac x a < \dfrac \pi 2 \\
-\dfrac 1 x \arccsc \dfrac x a + \dfrac {\sqrt{x^2 - a^2} } {a x} + C & : -\dfrac \pi 2 < \arccsc \dfrac x a < 0 \\
\end{cases}$
== 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:
First let $\arccsc \dfrac x a$ be in the interval $\openint 0 {\dfrac \pi 2}$.
Then:
Similarly, let $\arccsc \dfrac x a$ be in the interval $\openint {-\dfrac \pi 2} 0$.
Then:
== Also see ==
* Primitive of $\dfrac 1 {x^2} \arcsin \dfrac x a$
* Primitive of $\dfrac 1 {x^2} \arccos \dfrac x a$
* Primitive of $\dfrac 1 {x^2} \arctan \dfrac x a$
* Primitive of $\dfrac 1 {x^2} \arccot \dfrac x a$
* Primitive of $\dfrac 1 {x^2} \arcsec \dfrac x a$
== Sources ==
* : $\S 14$: Integrals involving Inverse Trigonometric Functions: $14.502$
Category:Primitives involving Inverse Cosecant Function
Primitive of Arccosine Function

== Theorem ==
:$\ds \int \arccos x \rd x = x \arccos x - \sqrt {1 - x^2} + C$
=== Corollary ===
== Proof 1 ==
== Proof 2 ==
== Also see ==
* Primitive of $\arcsin x$
* Primitive of $\arctan x$
* Primitive of $\arccot x$
== Sources ==
* : $6$. Integral Calculus: Appendix: Table of Fundamental Standard Integrals
* : Appendix: Table $2$: Integrals
* : Appendix: Table $2$: Integrals
Category:Primitive of Arccosine Function
Category:Primitives involving Inverse Cosine Function
Primitive of Arccosine 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:
Category:Primitive of Arccosine Function
Primitive of Arccosine of a x

== Theorem ==
:$\ds \int \arccos a x \rd x = x \arccos a x - \dfrac 1 a \sqrt {1 - a^2 x^2} + C$
== Proof ==
== Also see ==
* Primitive of $\arcsin a x$
* Primitive of $\arctan a x$
== Sources ==
* : Back endpapers: A Brief Table of Integrals: $97$.
Category:Primitives involving Inverse Cosine Function
Primitive of Arccosine of x over a

== Theorem ==
:$\ds \int \arccos \frac x a \rd x = x \arccos \frac x a - \sqrt {a^2 - x^2} + C$
== Proof 1 ==
== Proof 2 ==
== Proof 3 ==
== Also see ==
* Primitive of $\arcsin \dfrac x a$
* Primitive of $\arctan \dfrac x a$
* Primitive of $\arccot \dfrac x a$
* Primitive of $\arcsec \dfrac x a$
* Primitive of $\arccsc \dfrac x a$
== Sources ==
* : $\S 14$: Integrals involving Inverse Trigonometric Functions: $14.477$
* : Appendix $2$: Table of derivatives and integrals of common functions: Inverse trigonometric functions
Category:Primitive of Arccosine of x over a
Category:Primitive of Arccosine Function
Category:Primitives involving Inverse Cosine Function
Primitive of Arccosine of x over a/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:
Category:Primitive of Arccosine of x over a
Primitive of Arccosine of x over a over x

== Theorem ==
:$\ds \int \dfrac 1 x \arccos \frac x a \rd x = \frac \pi 2 \ln \size x - \int \frac {\arcsin \frac x a \rd x} x + C$
== Proof ==
== Also see ==
* Primitive of $\dfrac 1 x \arcsin \dfrac x a$
* Primitive of $\dfrac 1 x \arctan \dfrac x a$
* Primitive of $\dfrac 1 x \arccot \dfrac x a$
* Primitive of $\dfrac 1 x \arcsec \dfrac x a$
* Primitive of $\dfrac 1 x \arccsc \dfrac x a$
== Sources ==
* : $\S 14$: Integrals involving Inverse Trigonometric Functions: $14.480$
Category:Primitives involving Inverse Cosine Function
Primitive of Arccosine of x over a over x squared

== Theorem ==
:$\ds \int \frac 1 {x^2} \arccos \frac x a \rd x = -\frac 1 x \arccos \frac x a + \frac 1 a \map \ln {\frac {a + \sqrt {a^2 - x^2} } x} + C$
== 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 $\dfrac 1 {x^2} \arcsin \dfrac x a$
* Primitive of $\dfrac 1 {x^2} \arctan \dfrac x a$
* Primitive of $\dfrac 1 {x^2} \arccot \dfrac x a$
* Primitive of $\dfrac 1 {x^2} \arcsec \dfrac x a$
* Primitive of $\dfrac 1 {x^2} \arccsc \dfrac x a$
== Sources ==
* : $\S 14$: Integrals involving Inverse Trigonometric Functions: $14.481$
Category:Primitives involving Inverse Cosine Function
Primitive of Arccotangent Function

== Theorem ==
:$\ds \int \arccot x \rd x = x \arccot x + \frac {\map \ln {x^2 + 1} } 2 + C$
=== Corollary ===
== Proof 1 ==
== Proof 2 ==
== Also presented as ==
== Also see ==
* Primitive of $\arcsin x$
* Primitive of $\arccos x$
* Primitive of $\arctan x$
== Sources ==
* : Appendix: Table $2$: Integrals
* : Appendix: Table $2$: Integrals
Category:Primitive of Arccotangent Function
Category:Primitives involving Inverse Cotangent Function
Primitive of Arccotangent Function/Also presented as

== Primitive of Arccotangent Function: Also presented as ==
This result can also be presented as:
:$\ds \int \arccot x \rd x = x \arccot x + \ln \sqrt {x^2 + 1} + C$
== Proof ==
From Primitive of $\arccot x$:
:$\ds \int \arccot x \rd x = x \arccot x + \dfrac 1 2 \map \ln {x^2 + 1} + C$
The result follows by observing:
:$\dfrac 1 2 \log x = \log \sqrt x$
from Logarithm of Power.
== Sources ==
* : $6$. Integral Calculus: Appendix: Table of Fundamental Standard Integrals
Category:Primitives involving Inverse Cotangent Function
Primitive of Arccotangent 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:
Category:Primitive of Arccotangent Function
Primitive of Arccotangent of x over a

== Theorem ==
:$\ds \int \arccot \frac x a \rd x = x \arccot \frac x a + \frac a 2 \map \ln {x^2 + a^2} + C$
== Proof 1 ==
== Proof 2 ==
== Proof 3 ==
== Also see ==
* Primitive of $\arcsin \dfrac x a$
* Primitive of $\arccos \dfrac x a$
* Primitive of $\arctan \dfrac x a$
* Primitive of $\arcsec \dfrac x a$
* Primitive of $\arccsc \dfrac x a$
== Sources ==
* : $\S 14$: Integrals involving Inverse Trigonometric Functions: $14.488$
* : Appendix $2$: Table of derivatives and integrals of common functions: Inverse trigonometric functions
Category:Primitive of Arccotangent of x over a
Category:Primitive of Arccotangent Function
Category:Primitives involving Inverse Cotangent Function
Primitive of Arccotangent of x over a/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:
Category:Primitive of Arccotangent of x over a
Primitive of Arccotangent of x over a over x

== Theorem ==
:$\ds \int \dfrac 1 x \arccot \frac x a \rd x = \frac \pi 2 \ln \size x - \int \dfrac 1 x \arctan \frac x a \rd x$
== Proof ==
== Also see ==
* Primitive of $\dfrac 1 x \arcsin \dfrac x a$
* Primitive of $\dfrac 1 x \arccos \dfrac x a$
* Primitive of $\dfrac 1 x \arctan \dfrac x a$
* Primitive of $\dfrac 1 x \arcsec \dfrac x a$
* Primitive of $\dfrac 1 x \arccsc \dfrac x a$
== Sources ==
* : $\S 14$: Integrals involving Inverse Trigonometric Functions: $14.491$
Category:Primitives involving Inverse Cotangent Function
Primitive of Arccotangent of x over a over x squared

== Theorem ==
:$\ds \int \frac 1 {x^2} \arccot \frac x a \rd x = -\frac 1 x \arccot \frac x a + \frac 1 {2 a} \map \ln {\frac {x^2 + a^2} {x^2} } + C$
== 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 $\dfrac 1 {x^2} \arcsin \dfrac x a$
* Primitive of $\dfrac 1 {x^2} \arccos \dfrac x a$
* Primitive of $\dfrac 1 {x^2} \arctan \dfrac x a$
* Primitive of $\dfrac 1 {x^2} \arcsec \dfrac x a$
* Primitive of $\dfrac 1 {x^2} \arccsc \dfrac x a$
== Sources ==
* : $\S 14$: Integrals involving Inverse Trigonometric Functions: $14.492$
Category:Primitives involving Inverse Cotangent Function
Primitive of Arcsecant of x over a

== Theorem ==
=== Formulation 1 ===
=== Formulation 2 ===
== Also see ==
* Primitive of $\arcsin \dfrac x a$
* Primitive of $\arccos \dfrac x a$
* Primitive of $\arctan \dfrac x a$
* Primitive of $\arccot \dfrac x a$
* Primitive of $\arccsc \dfrac x a$
Category:Primitives involving Inverse Secant Function
Primitive of Arcsecant of x over a/Formulation 1

== Theorem ==
:$\ds \int \arcsec \frac x a \rd x = \begin {cases}
x \arcsec \dfrac x a - a \map \ln {x + \sqrt {x^2 - a^2} } + C & : 0 < \arcsec \dfrac x a < \dfrac \pi 2 \\
x \arcsec \dfrac x a + a \map \ln {x + \sqrt {x^2 - a^2} } + C & : \dfrac \pi 2 < \arcsec \dfrac x a < \pi \\
\end {cases}$
== 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:
First let $\arcsec \dfrac x a$ be in the interval $\openint 0 {\dfrac \pi 2}$.
Then:
Similarly, let $\arcsec \dfrac x a$ be in the interval $\openint {\dfrac \pi 2} \pi$.
Then:
== Also see ==
* Primitive of $\arcsin \dfrac x a$
* Primitive of $\arccos \dfrac x a$
* Primitive of $\arctan \dfrac x a$
* Primitive of $\arccot \dfrac x a$
* Primitive of $\arccsc \dfrac x a$
== Sources ==
* : $\S 14$: Integrals involving Inverse Trigonometric Functions: $14.493$
Category:Primitives involving Inverse Secant Function
Primitive of Arcsecant of x over a/Formulation 2

== Theorem ==
:$\ds \int \arcsec \frac x a \rd x = x \arcsec \frac x a - a \ln \size {x + \sqrt {x^2 - a^2} } + C$
for $x^2 > 1$.
$\arcsec \dfrac x a$ is undefined on the real numbers for $x^2 < 1$.
== 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:
We then have:
Let $x > 1$.
Then:
Similarly, let $x < -1$.
Then: 
== Sources ==
* : Appendix $2$: Table of derivatives and integrals of common functions: Inverse trigonometric functions
Category:Primitives involving Inverse Secant Function
Primitive of Arcsecant of x over a over x

== Theorem ==
<onlyinclude>
== Proof ==
== Also see ==
* Primitive of $\dfrac 1 x \arcsin \dfrac x a$
* Primitive of $\dfrac 1 x \arccos \dfrac x a$
* Primitive of $\dfrac 1 x \arctan \dfrac x a$
* Primitive of $\dfrac 1 x \arccot \dfrac x a$
* Primitive of $\dfrac 1 x \arccsc \dfrac x a$
== Sources ==
* : $\S 14$: Integrals involving Inverse Trigonometric Functions: $14.496$
Category:Primitives involving Inverse Secant Function
Primitive of Arcsecant of x over a over x squared

== Theorem ==
:$\ds \int \dfrac 1 {x^2} \arcsec \frac x a \rd x = \begin {cases}
-\dfrac 1 x \arcsec \dfrac x a + \dfrac {\sqrt {x^2 - a^2} } {a x} + C & : 0 < \arcsec \dfrac x a < \dfrac \pi 2 \\
-\dfrac 1 x \arcsec \dfrac x a - \dfrac {\sqrt {x^2 - a^2} } {a x} + C & : \dfrac \pi 2 < \arcsec \dfrac x a < \pi \\
\end {cases}$
== 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:
First let $\arcsec \dfrac x a$ be in the interval $\openint 0 {\dfrac \pi 2}$.
Then:
Similarly, let $\arcsec \dfrac x a$ be in the interval $\openint {\dfrac \pi 2} \pi$.
Then:
== Also see ==
* Primitive of $\dfrac 1 {x^2} \arcsin \dfrac x a$
* Primitive of $\dfrac 1 {x^2} \arccos \dfrac x a$
* Primitive of $\dfrac 1 {x^2} \arctan \dfrac x a$
* Primitive of $\dfrac 1 {x^2} \arccot \dfrac x a$
* Primitive of $\dfrac 1 {x^2} \arccsc \dfrac x a$
== Sources ==
* : $\S 14$: Integrals involving Inverse Trigonometric Functions: $14.497$
Category:Primitives involving Inverse Secant Function
Primitive of Arcsine Function

== Theorem ==
:$\ds \int \arcsin x \rd x = x \arcsin x + \sqrt {1 - x^2} + C$
=== Corollary ===
== Proof 1 ==
== Proof 2 ==
== Also see ==
* Primitive of $\arccos x$
* Primitive of $\arctan x$
* Primitive of $\arccot x$
== Sources ==
* : $6$. Integral Calculus: Appendix: Table of Fundamental Standard Integrals
* : Appendix: Table $2$: Integrals
* : Appendix: Table $2$: Integrals
Category:Primitive of Arcsine Function
Category:Primitives involving Inverse Sine Function
Primitive of Arcsine 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:
Category:Primitive of Arcsine Function
Primitive of Arcsine of a x

== Theorem ==
:$\ds \int \arcsin a x \rd x = x \arcsin a x + \dfrac 1 a \sqrt {1 - a^2 x^2} + C$
== Proof ==
== Also see ==
* Primitive of $\arccos a x$
* Primitive of $\arctan a x$
== Sources ==
* : Back endpapers: A Brief Table of Integrals: $96$.
Category:Primitives involving Inverse Sine Function
Primitive of Arcsine of x over a

== Theorem ==
:$\ds \int \arcsin \frac x a \rd x = x \arcsin \frac x a + \sqrt {a^2 - x^2} + C$
== Proof 1 ==
== Proof 2 ==
== Proof 3 ==
== Also see ==
* Primitive of $\arccos \dfrac x a$
* Primitive of $\arctan \dfrac x a$
* Primitive of $\arccot \dfrac x a$
* Primitive of $\arcsec \dfrac x a$
* Primitive of $\arccsc \dfrac x a$
== Sources ==
* : $\S 14$: Integrals involving Inverse Trigonometric Functions: $14.471$
* : Appendix $2$: Table of derivatives and integrals of common functions: Inverse trigonometric functions
* : $\S 17$: Tables of Special Indefinite Integrals: $(24)$ Integrals Involving Inverse Trigonometric Functions: $17.24.1.$
Category:Primitive of Arcsine of x over a
Category:Primitive of Arcsine Function
Category:Primitives involving Inverse Sine Function
Primitive of Arcsine of x over a/Proof 3

== 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:
Category:Primitive of Arcsine of x over a
Primitive of Arcsine of x over a over x

== Theorem ==
== Proof ==
== Also see ==
* Primitive of $\dfrac 1 x \arccos \dfrac x a$
* Primitive of $\dfrac 1 x \arctan \dfrac x a$
* Primitive of $\dfrac 1 x \arccot \dfrac x a$
* Primitive of $\dfrac 1 x \arcsec \dfrac x a$
* Primitive of $\dfrac 1 x \arccsc \dfrac x a$
== Sources ==
* : $\S 14$: Integrals involving Inverse Trigonometric Functions: $14.474$
Category:Primitives involving Inverse Sine Function
Primitive of Arcsine of x over a over x squared

== Theorem ==
:$\ds \int \dfrac 1 {x^2} \arcsin \frac x a \rd x = -\frac 1 x \arcsin \frac x a - \frac 1 a \map \ln {\frac {a + \sqrt {a^2 - x^2} } x} + C$
== 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 $\dfrac 1 {x^2} \arccos \dfrac x a$
* Primitive of $\dfrac 1 {x^2} \arctan \dfrac x a$
* Primitive of $\dfrac 1 {x^2} \arccot \dfrac x a$
* Primitive of $\dfrac 1 {x^2} \arcsec \dfrac x a$
* Primitive of $\dfrac 1 {x^2} \arccsc \dfrac x a$
== Sources ==
* : $\S 14$: Integrals involving Inverse Trigonometric Functions: $14.475$
Category:Primitives involving Inverse Sine Function
Primitive of Arctangent Function

== Theorem ==
:$\ds \int \arctan x \rd x = x \arctan x - \frac {\map \ln {x^2 + 1} } 2 + C$
=== Corollary ===
== Proof 1 ==
== Proof 2 ==
== Also presented as ==
== Also see ==
* Primitive of $\arcsin x$
* Primitive of $\arccos x$
* Primitive of $\arccot x$
== Sources ==
* 
* : Appendix: Table $2$: Integrals
* : Appendix: Table $2$: Integrals
Category:Primitive of Arctangent Function
Category:Primitives involving Inverse Tangent Function
Primitive of Arctangent Function/Also presented as

== Primitive of Arctangent Function: Also presented as ==
This result can also be presented as:
:$\ds \int \arctan x \rd x = x \arctan x - \ln \sqrt {x^2 + 1} + C$
== Proof ==
From Primitive of $\arccot x$:
:$\ds \int \arctan x \rd x = x \arctan x - \frac {\map \ln {x^2 + 1} } 2 + C$
The result follows by observing:
:$\dfrac 1 2 \log x = \log \sqrt x$
from Logarithm of Power.
== Sources ==
* : $6$. Integral Calculus: Appendix: Table of Fundamental Standard Integrals
Category:Primitives involving Inverse Tangent Function
Primitive of Arctangent 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:
Category:Primitive of Arctangent Function
Primitive of Arctangent of a x

== Theorem ==
:$\ds \int \arctan a x \rd x = x \arctan a x - \dfrac 1 {2 a} \map \ln {1 + a^2 x^2} + C$
== Proof ==
== Also see ==
* Primitive of $\arcsin a x$
* Primitive of $\arccos a x$
== Sources ==
* : Back endpapers: A Brief Table of Integrals: $98$.
Category:Primitives involving Inverse Tangent Function
Primitive of Arctangent of x over a

== Theorem ==
:$\ds \int \arctan \frac x a \rd x = x \arctan \frac x a - \frac a 2 \map \ln {x^2 + a^2} + C$
== Proof 1 ==
== Proof 2 ==
== Proof 3 ==
== Also see ==
* Primitive of $\arcsin \dfrac x a$
* Primitive of $\arccos \dfrac x a$
* Primitive of $\arccot \dfrac x a$
* Primitive of $\arcsec \dfrac x a$
* Primitive of $\arccsc \dfrac x a$
== Sources ==
* : $\S 14$: Integrals involving Inverse Trigonometric Functions: $14.483$
* : Appendix $2$: Table of derivatives and integrals of common functions: Inverse trigonometric functions
Category:Primitive of Arctangent of x over a
Category:Primitive of Arctangent Function
Category:Primitives involving Inverse Tangent Function
Primitive of Arctangent of x over a/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:
Category:Primitive of Arctangent of x over a
Primitive of Arctangent of x over a over x

== Theorem ==
== Proof ==
== Also see ==
* Primitive of $\dfrac 1 x \arcsin \dfrac x a$
* Primitive of $\dfrac 1 x \arccos \dfrac x a$
* Primitive of $\dfrac 1 x \arccot \dfrac x a$
* Primitive of $\dfrac 1 x \arcsec \dfrac x a$
* Primitive of $\dfrac 1 x \arccsc \dfrac x a$
== Sources ==
* : $\S 14$: Integrals involving Inverse Trigonometric Functions: $14.486$
Category:Primitives involving Inverse Tangent Function
Primitive of Arctangent of x over a over x squared

== Theorem ==
:$\ds \int \frac 1 {x^2} \arctan \frac x a \rd x = -\frac 1 x \arctan \frac x a - \frac 1 {2 a} \map \ln {\frac {x^2 + a^2} {x^2} } + C$
== 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 $\dfrac 1 {x^2} \arcsin \dfrac x a$
* Primitive of $\dfrac 1 {x^2} \arccos \dfrac x a$
* Primitive of $\dfrac 1 {x^2} \arccot \dfrac x a$
* Primitive of $\dfrac 1 {x^2} \arcsec \dfrac x a$
* Primitive of $\dfrac 1 {x^2} \arccsc \dfrac x a$
== Sources ==
* : $\S 14$: Integrals involving Inverse Trigonometric Functions: $14.487$
Category:Primitives involving Inverse Tangent Function
Primitive of Composite Function

== Theorem ==
Let $f$ and $g$ be a real functions which are integrable.
Let the composite function $g \circ f$ also be integrable.
Then:
where $u = \map f x$.
=== Corollary ===
== Proof ==
== Sources ==
* : $\S 14$: General Rules of Integration: $14.6$
* : $\S 16$: Indefinite Integrals: General Rules of Integration: $16.6.$
Category:Primitive of Composite Function
Category:Primitives
Primitive of Constant

== Theorem ==
Let $c$ be a constant.
:$\ds \int c \rd x = c x + C$ where $C$ is an arbitrary constant.
== Proof ==
Let:
:$\ds \map F x = \int c \rd x$
From the definition of primitive:
:$\map {F'} x = c$
From Derivative of Function of Constant Multiple:
:$\map {\dfrac \d {\d x} } {c x} = c$
From Primitives which Differ by Constant:
:$\map {\dfrac \d {\d x} } {c x + C} = c$
Hence the result.
== Sources ==
* : Part $\text {III}$: Derivatives and Integrals: Chapter $18$: Integration in Elementary Terms
* : $\S 14$: Indefinite Integrals: General Rules of Integration: $14.1$
* : $6$. Integral Calculus: Appendix: Table of Fundamental Standard Integrals
* : $\S 16$: Indefinite Integrals: General Rules of Integration: $16.1.$
Category:Primitives
Primitive of Constant Multiple of Function

== Theorem ==
Let $f$ be a real function which is integrable.
Let $c$ be a constant.
Then:
:$\ds \int c \map f x \rd x = c \int \map f x \rd x$
== Proof 1 ==
== Proof 2 ==
== Examples ==
== Sources ==
* : Chapter $10$: Integration: $10.4$. Standard integrals: General Rules: $\text {II}$.
* : Part $\text {III}$: Derivatives and Integrals: Chapter $18$: Integration in Elementary Terms
* : $\S 14$: General Rules of Integration: $14.2$
* : Appendix $\text I$: Table of Indefinite Integrals $1$.
* : Chapter $25$: Fundamental Integration Formulas: $3$.
* : $\S 16$: Indefinite Integrals: General Rules of Integration: $16.2.$
Category:Primitive of Constant Multiple of Function
Category:Primitives
Primitive of Constant Multiple of Function/Examples/2 cos x

== Example of Use of Primitive of Constant Multiple of Function ==
:$\ds \int 2 \cos x \rd x = 2 \sin x + C$
== Proof ==
== Sources ==
* : Chapter $\text {III}$: Integration: Three rules for integration: $\text {I}$
Category:Examples of Use of Primitive of Constant Multiple of Function
Primitive of Constant Multiple of Function/Proof 1

== Theorem ==
== Proof ==
From Linear Combination of Primitives:
:$\ds \int \paren {\lambda \map f x + \mu \map g x} \rd x = \lambda \int \map f x \rd x + \mu \int \map g x \rd x$
The result follows by setting $\lambda = c$ and $\mu = 0$.
Category:Primitive of Constant Multiple of Function
Primitive of Constant Multiple of Function/Proof 2

== Theorem ==
== Proof ==
From Derivative of Constant Multiple:
:$\map {\dfrac \d {\d x} } {c \map f x} = c \map {\dfrac \d {\d x} } {\map f x}$
The result follows from the definition of primitive.
== Sources ==
* : Chapter $\text {III}$: Integration: Three rules for integration: $\text {I}$
* 
Category:Primitive of Constant Multiple of Function
Primitive of Cosecant Function

== Theorem ==
=== Tangent Form ===
=== Cosecant plus Cotangent Form ===
=== Cosecant minus Cotangent Form ===
== Also presented as ==
== Also see ==
* Primitive of Sine Function
* Primitive of Cosine Function
* Primitive of Tangent Function
* Primitive of Cotangent Function
* Primitive of Secant Function
* Primitive of $\dfrac 1 {\sin a x}$
* Primitive of $\dfrac 1 {\cos a x}$
== Sources ==
* : $\text {II}$. 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: $8$
* : $\S 16$: Indefinite Integrals: General Rules of Integration: $16.16.$
* : Appendix $7$: Integrals
* : Appendix $8$: Integrals
Category:Primitive of Cosecant Function
Category:Primitives of Trigonometric Functions
Category:Primitives involving Cosecant Function
Category:Cosecant Function
Primitive of Cosecant Function/Cosecant minus 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 ==
* : $\text {II}$. Calculus: Integration
* : Chapter $10$: Integration: $10.4$. Standard integrals: Other Standard Results: $\text {(xix)}$
* : $\S 14$: General Rules of Integration: $14.16$
* : Chapter $25$: Fundamental Integration Formulas: $13$.
* : Chapter $5$. Integrals: Integrals of Special Functions: $8$
* : 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: $16.16.$
* : Appendix $7$: Integrals
* : Appendix $8$: Integrals
Category:Primitive of Cosecant Function