nyuuzyou/wikis huggingface dataset:
Primitive of Tangent Function/Cosine Form

== Theorem ==
:$\ds \int \tan x \rd x = -\ln \size {\cos x} + C$
where $\cos x \ne 0$.
== Proof ==
== Also see ==
* Primitive of $\tan x$: Secant Form
== Sources ==
* : Chapter $\text {III}$: Integration: Integration
* : $\S 14$: General Rules of Integration: $14.13$
* : Chapter $5$. Integrals: Integrals of Special Functions: $5$
* : $6$. 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: $16.13.$
* : Appendix $7$: Integrals
* : Appendix $8$: Integrals
Category:Primitive of Tangent Function
Primitive of Tangent Function/Secant Form

== Theorem ==
:$\ds \int \tan x \rd x = \ln \size {\sec x} + C$
where $\sec x$ is defined.
== Proof ==
== Also see ==
* Primitive of $\tan x$: Cosine Form
== Sources ==
* : $\text {II}$. Calculus: Integration
* : Chapter $10$: Integration: $10.4$. Standard integrals: Other Standard Results: $\text {(xvii)}$
* : $\S 14$: General Rules of Integration: $14.13$
* : Chapter $25$: Fundamental Integration Formulas: $10$.
* : Chapter $5$. Integrals: Integrals of Special Functions: $5$
* : $\S 16$: Indefinite Integrals: General Rules of Integration: $16.13.$
* : Appendix $7$: Integrals
* : Appendix $8$: Integrals
Category:Primitive of Tangent Function
Primitive of Tangent of a x

== Theorem ==
=== Secant Form ===
=== Cosine Form ===
== Examples ==
== Also see ==
* Primitive of $\sin a x$
* Primitive of $\cos a x$
* Primitive of $\cot a x$
* Primitive of $\sec a x$
* Primitive of $\csc a x$
Category:Primitive of Tangent of a x
Category:Primitive of Tangent Function
Category:Primitives involving Tangent Function
Primitive of Tangent of a x/Cosine Form

== Theorem ==
:$\ds \int \tan a x \rd x = \frac {-\ln \size {\cos a x} } a + C$
== Proof ==
== Also see ==
* Primitive of $\tan a x$: Secant Form
== Sources ==
* : $\S 14$: Integrals involving $\tan a x$: $14.429$
* : Back endpapers: A Brief Table of Integrals: $82$.
* : $\S 17$: Tables of Special Indefinite Integrals: $(20)$ Integrals Involving $\tan a x$: $17.20.1.$
Category:Primitive of Tangent of a x
Primitive of Tangent of a x/Examples/2 x

== Example of Use of Primitive of $\tan a x$ ==
:$\ds \int \tan 3 x \rd x = \frac 1 2 \ln \size {\sec 2 x} + C$
== Proof ==
From Primitive of $\tan a x$: Secant Form:
The result follows by setting $a = 2$.
== Proof ==
* : $\text {II}$. Calculus: Exercises $\text {XIV}$: $2$.
Category:Examples of Use of Primitive of Tangent of a x
Primitive of Tangent of a x/Secant Form

== Theorem ==
:$\ds \int \tan a x \rd x = \frac {\ln \size {\sec a x} } a + C$
== Proof ==
== Also see ==
* Primitive of $\tan a x$: Cosine Form
== Sources ==
* : $\S 14$: Integrals involving $\tan a x$: $14.429$
* : $\S 17$: Tables of Special Indefinite Integrals: $(20)$ Integrals Involving $\tan a x$: $17.20.1.$
Category:Primitive of Tangent of a x
Primitive of Tangent of a x over x

== Theorem ==
:$\ds \int \frac {\tan a x} x \rd x = a x + \frac {\paren {a x}^3} 9 + \frac {2 \paren {a x}^5} {75} + \cdots + \frac {\paren {-1}^{n - 1} 2^{2 n} \paren {2^{2 n} - 1} B_{2 n} \paren {a x}^{2 n - 1} } {\paren {2 n - 1} \paren {2 n}!} + \cdots + C$
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 $\dfrac {\cot a x} x$
* Primitive of $\dfrac {\sec a x} x$
* Primitive of $\dfrac {\csc a x} x$
== Sources ==
* : $\S 14$: Integrals involving $\tan a x$: $14.436$
Category:Primitives involving Tangent Function
Primitive of a x + b over p x + q

== Theorem ==
:$\ds \int \frac {a x + b} {p x + q} \rd x = \frac {a x} p + \frac {b p - a q} {p^2} \ln \size {p x + q} + C$
== Proof ==
== Sources ==
* : $\S 14$: Integrals involving $a x + b$ and $p x + q$: $14.111$
* : $\S 17$: Tables of Special Indefinite Integrals: $(3)$ Integrals Involving $a x + b$ and $p x + q$: $17.3.7.$
Category:Primitives involving a x + b and p x + q
Primitive of exp (-x^2) has no Solution in Elementary Functions

== Theorem ==
The primitive:
:$\ds \int \map \exp {-x^2} \rd x$
cannot be expressed in terms of a finite number of elementary functions.
== Proof ==
== Historical Note ==
== Sources ==
* : Chapter $\text {A}.29$: Liouville ()
Category:Primitives involving Exponential Function
Primitive of exp (-x^2) has no Solution in Elementary Functions/Historical Note

== Historical Note on Primitive of $\map \exp {-x^2}$ has no Solution in Elementary Functions ==
The proof that $\ds \int \map \exp {-x^2} \rd x$ cannot be expressed with a finite number of elementary functions was first proved by .
== Sources ==
* : Chapter $\text {A}.29$: Liouville ()
Category:Historical Notes
Primitive of exp x over x has no Solution in Elementary Functions

== Theorem ==
The primitive:
:$\ds \int \frac {e^{a x} \rd x} x$
cannot be expressed in terms of a finite number of elementary functions.
== Proof ==
== Also see ==
* Primitive of $\dfrac {e^{a x} } x$
== Historical Note ==
== Sources ==
* : Chapter $1$: Linear Differential Equations with Constant Coefficients: $\S 1$. The first order equation: $\S 1.2$ The integrating factor: Example $2$
* : Chapter $\text {A}.29$: Liouville ()
Category:Primitives involving Exponential Function
Primitive of exp x over x has no Solution in Elementary Functions/Historical Note

== Historical Note on Primitive of $\dfrac {e^x} x$ has no Solution in Elementary Functions ==
The proof that $\ds \int \dfrac {e^x \rd x} x$ cannot be expressed with a finite number of elementary functions was first proved by .
== Sources ==
* : Chapter $\text {A}.29$: Liouville ()
Category:Historical Notes
Primitive of p x + q over Root of a x + b

== Theorem ==
:$\ds \int \frac {p x + q} {\sqrt {a x + b} } \rd x = \frac {2 \paren {a p x + 3 a q - 2 b p} } {3 a^2} \sqrt{a x + b}$
== Proof ==
Let:
Then:
== Sources ==
* : $\S 14$: Integrals involving $\sqrt {a x + b}$ and $p x + q$: $14.113$
* : $\S 17$: Tables of Special Indefinite Integrals: $(4)$ Integrals Involving $\sqrt {a x + b}$ and $p x + q$: $17.4.1.$
Category:Primitives involving Root of a x + b and p x + q
Primitive of p x + q over Root of a x squared plus 2 b x plus c

== Theorem ==
:$\ds \int \dfrac {p x + q} {\sqrt {a x^2 + 2 b x + c} } \rd x = \dfrac p a \sqrt {a x^2 + 2 b x + c} + \paren {q - \dfrac {p b} a} \int \dfrac {\d x} {\sqrt {a x^2 + 2 b x + c} }$
== Proof 1 ==
== Proof 2 ==
== Examples ==
Category:Primitive of p x + q over Root of a x squared plus 2 b x plus c
Category:Primitives involving Root of a x squared plus b x plus c
Primitive of p x + q over Root of a x squared plus 2 b x plus c/Examples

== Examples of Use of Primitive of $\dfrac {p x + q} {\sqrt {a x^ 2 + 2 b x + c} }$ ==
=== Primitive of $\dfrac {1 + x} {\sqrt {2 + 4 x - 3 x^2} }$ ===
Category:Primitive of p x + q over Root of a x squared plus 2 b x plus c
Primitive of p x + q over Root of a x squared plus 2 b x plus c/Examples/1 + x over Root 2 + 4 x - 3 x^2

== Example of Use of Primitive of $\dfrac {p x + q} {\sqrt {a x^ 2 + 2 b x + c} }$ ==
:$\ds \int \dfrac {1 + x} {\sqrt {2 + 4 x - 3 x^2} } \rd x = -\dfrac 1 3 \sqrt {2 + 4 x - 3 x^2} + \dfrac 5 {3 \sqrt 3} \map \arcsin {\dfrac {3 x - 2} {\sqrt {10} } } + C$
== Proof ==
== Sources ==
* : $\text {II}$. Calculus: Integration: Algebraic Integration: Example
Category:Primitive of p x + q over Root of a x squared plus 2 b x plus c
Primitive of p x + q over a x squared plus 2 b x plus c

== Theorem ==
:$\ds \int \dfrac {p x + q} {a x^2 + 2 b x + c} \rd x = \dfrac p {2 a} \ln \size {a x^2 + 2 b x + c} + \paren {q - \dfrac {p b} a} \int \dfrac {\d x} {a^2 + 2 b x + c} + C$
== Proof ==
== Examples ==
== Sources ==
* : $\text {II}$. Calculus: Integration: Algebraic Integration: Type $\text B$
Category:Primitive of p x + q over a x squared plus 2 b x plus c
Category:Primitives involving a x squared plus b x plus c
Primitive of p x + q over a x squared plus 2 b x plus c/Examples

== Examples of Use of Primitive of $\dfrac {p x + q} {a x^2 + 2 b x + c}$ ==
=== Primitive of $\dfrac {2 x + 1} {3 x^2 + 4 x + 2}$ ===
Category:Primitive of p x + q over a x squared plus 2 b x plus c
Primitive of p x + q over a x squared plus 2 b x plus c/Examples/2 x + 1 over 3 x^2 + 4 x + 2

== Example of Use of Primitive of $\dfrac {p x + q} {a x^2 + 2 b x + c}$ ==
:$\ds \int \dfrac {2 x + 1} {3 x^2 + 4 x + 2} \rd x = \dfrac 1 3 \ln \size {3 x^2 + 4 x + 2} - \dfrac 1 {3 \sqrt 2} \map \arctan {\dfrac {3 x + 2} {\sqrt 2} } + C$
== Proof 1 ==
== Proof 2 ==
Category:Primitive of p x + q over a x squared plus 2 b x plus c/Examples/2 x + 1 over 3 x^2 + 4 x + 2
Category:Primitive of p x + q over a x squared plus 2 b x plus c
Primitive of p x + q over a x squared plus 2 b x plus c/Examples/2 x + 1 over 3 x^2 + 4 x + 2/Proof 1

== Example of Use of Primitive of $\dfrac {p x + q} {a x^2 + 2 b x + c}$ ==
== Proof ==
Category:Primitive of p x + q over a x squared plus 2 b x plus c/Examples/2 x + 1 over 3 x^2 + 4 x + 2
Primitive of p x + q over a x squared plus 2 b x plus c/Examples/2 x + 1 over 3 x^2 + 4 x + 2/Proof 2

== Example of Use of Primitive of $\dfrac {p x + q} {a x^2 + 2 b x + c}$ ==
== Proof ==
== Sources ==
* : $\text {II}$. Calculus: Integration: Algebraic Integration: Type $\text B$.
Category:Primitive of p x + q over a x squared plus 2 b x plus c/Examples/2 x + 1 over 3 x^2 + 4 x + 2
Primitive of x by Arccosecant of x over a

== Theorem ==
:$\ds \int x \arccsc \frac x a \rd x = \begin{cases}
\dfrac {x^2} 2 \arccsc \dfrac x a + \dfrac {a \sqrt {x^2 - a^2} } 2 + C & : 0 < \arccsc \dfrac x a < \dfrac \pi 2 \\
\dfrac {x^2} 2 \arccsc \dfrac x a - \dfrac {a \sqrt {x^2 - a^2} } 2 + C & : -\dfrac \pi 2 < \arccsc \dfrac x a < 0 \\
\end{cases}$
== 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 $x \arcsin \dfrac x a$
* Primitive of $x \arccos \dfrac x a$
* Primitive of $x \arctan \dfrac x a$
* Primitive of $x \arccot \dfrac x a$
* Primitive of $x \arcsec \dfrac x a$
== Sources ==
* : $\S 14$: Integrals involving Inverse Trigonometric Functions: $14.499$
Category:Primitives involving Inverse Cosecant Function
Primitive of x by Arccosine of x over a

== Theorem ==
:$\ds \int x \arccos \frac x a \rd x = \paren {\frac {x^2} 2 - \frac {a^2} 4} \arccos \frac x a - \frac {x \sqrt {a^2 - x^2} } 4 + C$
== Proof 1 ==
== Proof 2 ==
== Also see ==
* Primitive of $x \arcsin \dfrac x a$
* Primitive of $x \arctan \dfrac x a$
* Primitive of $x \arccot \dfrac x a$
* Primitive of $x \arcsec \dfrac x a$
* Primitive of $x \arccsc \dfrac x a$
== Sources ==
* : $\S 14$: Integrals involving Inverse Trigonometric Functions: $14.478$
Category:Primitives involving Inverse Cosine Function
Category:Primitive of x by Arccosine of x over a
Primitive of x by 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 x by Arccosine of x over a
Primitive of x by Arccotangent of x over a

== Theorem ==
:$\ds \int x \arccot \frac x a \rd x = \frac {x^2 + a^2} 2 \arccot \frac x a + \frac {a x} 2 + C$
== Proof ==
<onlyinclude>
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 $x \arcsin \dfrac x a$
* Primitive of $x \arccos \dfrac x a$
* Primitive of $x \arctan \dfrac x a$
* Primitive of $x \arcsec \dfrac x a$
* Primitive of $x \arccsc \dfrac x a$
== Sources ==
* : $\S 14$: Integrals involving Inverse Trigonometric Functions: $14.489$
Category:Primitives involving Inverse Cotangent Function
Primitive of x by Arcsecant of x over a

== Theorem ==
:$\ds \int x \arcsec \frac x a \rd x = \begin{cases}
\dfrac {x^2} 2 \arcsec \dfrac x a - \dfrac {a \sqrt {x^2 - a^2} } 2 + C & : 0 < \arcsec \dfrac x a < \dfrac \pi 2 \\
\dfrac {x^2} 2 \arcsec \dfrac x a + \dfrac {a \sqrt {x^2 - a^2} } 2 + C & : \dfrac \pi 2 < \arcsec \dfrac x a < \pi \\
\end{cases}$
== 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 $x \arcsin \dfrac x a$
* Primitive of $x \arccos \dfrac x a$
* Primitive of $x \arctan \dfrac x a$
* Primitive of $x \arccot \dfrac x a$
* Primitive of $x \arccsc \dfrac x a$
== Sources ==
* : $\S 14$: Integrals involving Inverse Trigonometric Functions: $14.494$
Category:Primitives involving Inverse Secant Function
Primitive of x by Arcsine of x over a

== Theorem ==
:$\ds \int x \arcsin \frac x a \rd x = \paren {\frac {x^2} 2 - \frac {a^2} 4} \arcsin \frac x a + \frac {x \sqrt {a^2 - x^2} } 4 + C$
== Proof 1 ==
== Proof 2 ==
== Also see ==
* Primitive of $x \arccos \dfrac x a$
* Primitive of $x \arctan \dfrac x a$
* Primitive of $x \arccot \dfrac x a$
* Primitive of $x \arcsec \dfrac x a$
* Primitive of $x \arccsc \dfrac x a$
== Sources ==
* : $\S 14$: Integrals involving Inverse Trigonometric Functions: $14.472$
* : $\S 17$: Tables of Special Indefinite Integrals: $(24)$ Integrals Involving Inverse Trigonometric Functions: $17.24.2.$
Category:Primitives involving Inverse Sine Function
Category:Primitive of x by Arcsine of x over a
Primitive of x by Arcsine 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 x by Arcsine of x over a
Primitive of x by Arctangent of x

== Theorem ==
:$\ds \int x \arctan x \rd x = \frac {x^2 + 1} 2 \arctan x - \frac x 2 + C$
== Proof 1 ==
== Proof 2 ==
== Sources ==
* : $\text {II}$. Calculus: Exercises $\text {XIV}$: $24$.
Category:Primitive of x by Arctangent of x
Category:Primitives involving Inverse Tangent Function
Primitive of x by Arctangent of 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:
== Sources ==
* : $\text {II}$. Calculus: Integration: Integration by Parts: Example
Category:Primitive of x by Arctangent of x
Primitive of x by Arctangent of x over a

== Theorem ==
:$\ds \int x \arctan \frac x a \rd x = \frac {x^2 + a^2} 2 \arctan \frac x a - \frac {a x} 2 + C$
== 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 $x \arcsin \dfrac x a$
* Primitive of $x \arccos \dfrac x a$
* Primitive of $x \arccot \dfrac x a$
* Primitive of $x \arcsec \dfrac x a$
* Primitive of $x \arccsc \dfrac x a$
== Sources ==
* : $\S 14$: Integrals involving Inverse Trigonometric Functions: $14.484$
Category:Primitives involving Inverse Tangent Function
Primitive of x by Cosecant of a x

== Theorem ==
:$\ds \int x \csc a x \rd x = \frac 1 {a^2} \paren {a x + \frac {\paren {a x}^3} {18} + \frac {7 \paren {a x}^5} {1800} + \cdots + \frac {\paren {-1}^{n - 1} 2 \paren {2^{2 n - 1} - 1} B_{2n} \paren {a x}^{2 n + 1} } {\paren {2 n + 1}!} + \cdots} + C$
where $B_{2 n}$ is the $2 n$th Bernoulli number.
== Proof ==
 
== Also see ==
* Primitive of $x \sin a x$
* Primitive of $x \cos a x$
* Primitive of $x \tan a x$
* Primitive of $x \cot a x$
* Primitive of $x \sec a x$
== Sources ==
* : $\S 14$: Integrals involving $\csc a x$: $14.466$
Category:Primitives involving Cosecant Function
Primitive of x by Cosine of a x

== Theorem ==
:$\ds \int x \cos a x \rd x = \frac {\cos a x} {a^2} + \frac {x \sin a x} a + C$
where $C$ is an arbitrary constant.
== 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 $x \sin a x$
* Primitive of $x \tan a x$
* Primitive of $x \cot a x$
* Primitive of $x \sec a x$
* Primitive of $x \csc a x$
== Sources ==
* : $\S 14$: Integrals involving $\cos a x$: $14.370$
* : Back endpapers: A Brief Table of Integrals: $79$.
* : $\S 17$: Tables of Special Indefinite Integrals: $(18)$ Integrals Involving $\cos a x$: $17.18.2.$
Category:Primitives involving Cosine Function
Primitive of x by Cosine of x

== Theorem ==
:$\ds \int x \cos x \rd x = \cos x + x \sin x + C$
where $C$ is an arbitrary constant.
== Proof 1 ==
== Proof 2 ==
== Proof 3 ==
== Warning ==
Category:Primitive of x by Cosine of x
Category:Primitives involving Cosine Function
Primitive of x by Cosine of x/Proof 1

== Theorem ==
== Proof ==
From Primitive of $x \cos a x$:
:$\ds \int x \cos a x \rd x = \frac {\cos a x} {a^2} + \frac {x \sin a x} a + C$
The result follows on setting $a = 1$.
Category:Primitive of x by Cosine of x
Primitive of x by Cosine of 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:
== Sources ==
* 
* 
Category:Primitive of x by Cosine of x
Primitive of x by Cosine of x/Warning

== Primitive of $x \cos x$: Warning ==
Consider the primitive:
:$\ds \int x \cos x \rd x$
It is a mistake to use Primitive of Constant Multiple of Function here:
:$\ds \int x \cos x \rd x = x \int \cos x \rd x$
and thence to deduce:
:$\ds \int x \cos x \rd x = x \sin x + C$
because $x$ is not a constant in the expression $x \cos x$.
== Sources ==
* : Chapter $\text {III}$: Integration: Three rules for integration: $\text {I}$
Category:Primitive of x by Cosine of x
Primitive of x by Cotangent of a x

== Theorem ==
:$\ds \int x \cot a x \rd x = \frac 1 {a ^ 2} \paren {a x - \frac {\paren {a x}^3} 9 - \frac {\paren {a x}^5} {225} - \cdots + \frac {\paren {-1}^n 2^{2 n} B_{2 n} \paren {a x}^{2 n + 1} } {\paren {2 n + 1} !} + \cdots} + C$
where $B_{2 n}$ denotes the $2 n$th Bernoulli number.
== Proof ==
From Power Series Expansion for Cotangent Function:
== Also see ==
* Primitive of $x \sin a x$
* Primitive of $x \cos a x$
* Primitive of $x \tan a x$
* Primitive of $x \sec a x$
* Primitive of $x \csc a x$
== Sources ==
* : $\S 14$: Integrals involving $\cot a x$: $14.446$
Category:Primitives involving Cotangent Function
Primitive of x by Exponential of a x

== Theorem ==
:$\ds \int x e^{a x} \rd x = \frac {e^{a x} } a \paren {x - \frac 1 a} + C$
== 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 presented as ==
== Also see ==
* Primitive of $x^n e^{a x}$
* Primitive of $x^2 e^{a x}$
== Sources ==
* : $\S 14$: Integrals involving $e^{a x}$: $14.510$
* : $\S 17$: Tables of Special Indefinite Integrals: $(25)$ Integrals Involving $e^{a x}$: $17.25.2.$
Category:Primitive of x by Exponential of a x
Category:Primitives involving Exponential Function
Primitive of x by Exponential of a x/Also presented as

== Primitive of $x e^{a x} \rd x$: Also presented as ==
This result is also seen presented in the form:
:$\ds \int x e^{a x} \rd x = \frac {e^{a x} } {a^2} \paren {a x - 1} + C$
== Proof ==
== Sources ==
* : Back endpapers: A Brief Table of Integrals: $104$.
Category:Primitive of x by Exponential of a x
Primitive of x by Exponential of a x by Cosine of b x

== Theorem ==
:$\ds \int x e^{a x} \cos b x \rd x = \frac {x e^{a x} \paren {a \cos b x + b \sin b x} } {a^2 + b^2} - \frac {e^{a x} \paren {\paren {a^2 - b^2} \cos b x - 2 a b \sin b x} } {\paren {a^2 + b^2}^2} + C$
== 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 $x e^{a x} \sin b x$
== Sources ==
* : $\S 14$: Integrals involving $e^{a x}$: $14.521$
Category:Primitives involving Exponential Function
Category:Primitives involving Cosine Function
Primitive of x by Exponential of a x by Sine of b x

== Theorem ==
:$\ds \int x e^{a x} \sin b x \rd x = \frac {x e^{a x} \paren {a \sin b x - b \cos b x} } {a^2 + b^2} - \frac {e^{a x} \paren {\paren {a^2 - b^2} \sin b x - 2 a b \cos b x} } {\paren {a^2 + b^2}^2} + C$
== 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 $x e^{a x} \cos b x$
== Sources ==
* : $\S 14$: Integrals involving $e^{a x}$: $14.520$
Category:Primitives involving Exponential Function
Category:Primitives involving Sine Function
Primitive of x by Half Integer Power of a x squared plus b x plus c

== Theorem ==
Let $a \in \R_{\ne 0}$.
Then:
:$\ds \int x \paren {a x^2 + b x + c}^{n + \frac 1 2} \rd x = \frac {\paren {a x^2 + b x + c}^{n + \frac 3 2} } {a \paren {2 n + 3} } - \frac b {2 a} \int \paren {a x^2 + b x + c}^{n + \frac 1 2} \rd x$ 
== Proof ==
Let:
So:
== Sources ==
* : $\S 14$: Integrals involving $\sqrt {a x^2 + b x + c}$: $14.296$
Category:Primitives involving Root of a x squared plus b x plus c
Primitive of x by Hyperbolic Cosecant of a x

== Theorem ==
where $B_{2 n}$ denotes the $2 n$th Bernoulli number.
== Proof ==
 
== Also see ==
* Primitive of $x \sinh a x$
* Primitive of $x \cosh a x$
* Primitive of $x \tanh a x$
* Primitive of $x \coth a x$
* Primitive of $x \sech a x$
== Sources ==
* : $\S 14$: Integrals involving $\csch a x$: $14.641$
* : $\S 17$: Tables of Special Indefinite Integrals: $(33)$ Integrals Involving $\csch a x$: $17.33.4.$
Category:Primitives involving Hyperbolic Cosecant Function
Primitive of x by Hyperbolic Cosine of a x

== Theorem ==
:$\ds \int x \cosh a x \rd x = \frac {x \sinh a x} a - \frac {\cosh a x} {a^2} + C$
where $C$ is an arbitrary constant.
== Proof ==
With a view to expressing the primitive in the form:
:$\ds \int u \frac {\d v} {\d x} \rd x = u v - \int v \frac {\rd u} {\rd x} \rd x$
let:
and let:
Then:
== Also see ==
* Primitive of $x \sinh a x$
* Primitive of $x \tanh a x$
* Primitive of $x \coth a x$
* Primitive of $x \sech a x$
* Primitive of $x \csch a x$
== Sources ==
* : $\S 14$: Integrals involving $\sinh a x$: $14.563$
* : Back endpapers: A Brief Table of Integrals: $120$.
* : $\S 17$: Tables of Special Indefinite Integrals: $(28)$ Integrals Involving $\cosh a x$: $17.28.2.$
Category:Primitives involving Hyperbolic Cosine Function
Primitive of x by Hyperbolic Cotangent of a x

== Theorem ==
:$\ds \int x \coth a x \rd x = \frac 1 {a^2} \paren {a x + \frac {\paren {a x}^3} 9 - \frac {\paren {a x}^5} {225} + \cdots + \frac {2^{2 n} B_{2 n} \paren {a x}^{2 n + 1} } {\paren {2 n + 1}!} + \cdots} + C$
where $B_{2 n}$ denotes the $2 n$th Bernoulli number.
== Proof ==
 
== Also see ==
* Primitive of $x \sinh a x$
* Primitive of $x \cosh a x$
* Primitive of $x \tanh a x$
* Primitive of $x \sech a x$
* Primitive of $x \csch a x$
== Sources ==
* : $\S 14$: Integrals involving $\tanh a x$: $14.621$
* : $\S 17$: Tables of Special Indefinite Integrals: $(31)$ Integrals Involving $\coth a x$: $17.31.4.$
Category:Primitives involving Hyperbolic Cotangent Function
Primitive of x by Hyperbolic Secant of a x

== Theorem ==
where $E_{2 n}$ denotes the $2 n$th Euler number.
== Proof ==
 
== Also see ==
* Primitive of $x \sinh a x$
* Primitive of $x \cosh a x$
* Primitive of $x \tanh a x$
* Primitive of $x \coth a x$
* Primitive of $x \csch a x$
== Sources ==
* : $\S 14$: Integrals involving $\sech a x$: $14.631$
* : $\S 17$: Tables of Special Indefinite Integrals: $(32)$ Integrals Involving $\sech a x$: $17.32.4.$
Category:Primitives involving Hyperbolic Secant Function