nyuuzyou/wikis huggingface dataset:
Primitive of Exponential Function/Examples/1 - x

== Example of Use of Primitive of Exponential Function ==
:$\ds \int e^{1 - x} \rd x = -e^{1 - x} + C$
== Proof ==
From Primitive of Function of $a x + b$:
Hence:
== Proof ==
* : $\text {II}$. Calculus: Exercises $\text {XIV}$: $4$.
Category:Examples of Use of Primitive of Exponential Function
Primitive of Exponential Function/General Result

== Theorem ==
Let $a \in \R_{>0}$ be a constant such that $a \ne 1$.
Then:
:$\ds \int a^x \rd x = \frac {a^x} {\ln a} + C$
where $C$ is an arbitrary constant.
== Proof 1 ==
== Proof 2 ==
== Sources ==
* : $\text {II}$. Calculus: Integration
* : Chapter $10$: Integration: $10.4$. Standard integrals: Other Standard Results: $\text {(xii)}$
* : $\S 14$: General Rules of Integration: $14.10$
* : Front endpapers: A Brief Table of Integrals: $2$.
* : Appendix $\text I$: Table of Indefinite Integrals $9$.
* : $6$. Integral Calculus: Appendix: Table of Fundamental Standard Integrals
* : Appendix: Table $2$: Integrals
* : Appendix: Table $2$: Integrals
* : $\S 16$: Indefinite Integrals: General Rules of Integration: $16.10.$
* : Appendix $7$: Integrals
* : Appendix $8$: Integrals
Category:Primitive of Exponential Function
Primitive of Exponential Integral Function

== Theorem ==
Let $\Ei: \R_{>0} \to \R$ denote the exponential integral function:
:$\map \Ei x = \ds \int_{t \mathop = x}^{t \mathop \to +\infty} \frac {e^{-t} } t \rd t$
Then:
:$\ds \int \map \Ei x \rd x = x \map \Ei x - e^{-x} + C$
== Proof ==
By Derivative of Exponential Integral Function, we have: 
:$\ds \frac \d {\d x} \paren {\map \Ei x} = -\frac {e^{-x} } x$
So: 
Category:Primitives
Category:Exponential Integral Function
Primitive of Exponential of a x

== Theorem ==
:$\ds \int e^{a x} \rd x = \frac {e^{a x} } a + C$
where $a$ is a non-zero constant.
== Proof for Real Numbers ==
== Proof for Complex Numbers ==
== Sources ==
* : Chapter $\text {II}$: Integration of Elementary Functions: $\S 7$. Standard Integrals: $3$.
* : Chapter $10$: Integration: $10.4$. Standard integrals: Standard Forms: $\text {(iii)}$
* : $\S 14$: Integrals involving $e^{a x}$: $14.509$
* : Back endpapers: A Brief Table of Integrals: $102$.
* : Appendix $\text I$: Table of Indefinite Integrals $8$.
* : $9$ Sums and Integrals: $9.8$ Standard Integrals
* : Appendix: Table $2$: Integrals
* : Appendix: Table $2$: Integrals
* : $\S 17$: Tables of Special Indefinite Integrals: $(25)$ Integrals Involving $e^{a x}$: $17.25.1.$
* : Appendix $7$: Integrals
* : Appendix $8$: Integrals
Category:Primitives involving Exponential Function
Category:Primitive of Exponential of a x
Primitive of Exponential of a x by Cosine of b x

== Theorem ==
:$\ds \int e^{a x} \cos b x \rd x = \frac {e^{a x} \paren {a \cos b x + b \sin b x} } {a^2 + b^2} + C$
== Proof 1 ==
== Proof 2 ==
== Proof 3 ==
== Also see ==
* Primitive of $e^{a x} \sin b x$
== Sources ==
* : $\S 14$: Integrals involving $e^{a x}$: $14.519$
* : Back endpapers: A Brief Table of Integrals: $108$.
* : Appendix $7$: Integrals
* : Appendix $8$: Integrals
Category:Primitive of Exponential of a x by Cosine of b x
Category:Primitives involving Exponential Function
Category:Primitives involving Cosine Function
Primitive of Exponential of a x by Cosine of b x/Lemma

== Lemma for Primitive of $e^{a x} \cos b x$ ==
:$\ds \int e^{a x} \cos b x \rd x = \frac {e^{a x} \cos b x} a + \frac b a \int e^{a x} \sin b x \rd x$
== 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:
Category:Primitive of Exponential of a x by Cosine of b x
Primitive of Exponential of a x by Cosine of b x/Proof 3

== Theorem == 
== Proof ==
Let $a, b \in \R_{>0}$ be real constants.
Let $f_1$ and $f_2$ be the real functions defined as:
Let $\map \CC \R$ denote the space of continuous real-valued functions.
Let $\struct {\map {\CC^1} \R, +, \, \cdot \,}_\R$ denote the vector space of continuously differentiable real-valued functions.
Let $S = \span \set {f_1, f_2} \subset \map {\CC^1} \R$ be a vector space.
Let $D : S \to S$ be the derivative $x$.
From Differentiation of Exponential of a x by Cosine of b x and Exponential of a x by Sine of b x wrt x as Nonsingular Matrix, $D$ is expressible as:
:$\mathbf D = \begin {bmatrix} a & b \\ -b & a \end {bmatrix}$
and is nonsingular.
By Inverse of Matrix is Scalar Product of Adjugate by Reciprocal of Determinant:
:$\mathbf D^{-1} = \dfrac 1 {a^2 + b^2} \begin {bmatrix} a & -b \\ b & a \end {bmatrix}$
Then:
Application of $\mathbf D$ on both sides on the left and writing out explicitly in terms of $f_1$ and $f_2$ yields:
:$f_1 = \dfrac \d {\d x} \dfrac {a f_1 + b f_2} {a^2 + b^2}$
Integrating $x$:
:$\ds \int f_1 \rd x = \frac {a f_1 + b f_2} {a^2 + b^2} + C$
where $C$ is an arbitrary constant.
Substitute definitions of $f_1$ and $f_2$ to get the desired result.
== Sources ==
* : Chapter $\S 2.1$: Continuous and linear maps. Linear transformations
Category:Primitive of Exponential of a x by Cosine of b x
Primitive of Exponential of a x by Hyperbolic Cosine of b x

== Theorem ==
=== Exponential Form ===
=== Hyperbolic Form ===
== Also see ==
* Primitive of $e^{a x} \sinh b x$
Category:Primitive of Exponential of a x by Hyperbolic Cosine of b x
Category:Primitives involving Exponential Function
Category:Primitives involving Hyperbolic Cosine Function
Primitive of Exponential of a x by Hyperbolic Cosine of b x/Exponential Form

== Theorem ==
:$\ds \int e^{a x} \cosh b x \rd x = \frac {e^{a x} } 2 \paren {\frac {e^{b x} } {a + b} + \frac {e^{-b x} } {a - b} } + C$
for $a^2 \ne b^2$.
== Proof ==
== Also see ==
* Primitive of $e^{a x} \sinh b x$
== Sources ==
* : Back endpapers: A Brief Table of Integrals: $138$.
Category:Primitive of Exponential of a x by Hyperbolic Cosine of b x
Primitive of Exponential of a x by Hyperbolic Cosine of b x/Hyperbolic Form

== Theorem ==
:$\ds \int e^{a x} \cosh b x \rd x = \frac {e^{a x} \paren {a \cosh b x + b \sinh b x} } {a^2 - b^2} + C$
for $a^2 \ne b^2$.
== Proof ==
== Also see ==
* Primitive of $e^{a x} \sinh b x$
Category:Primitive of Exponential of a x by Hyperbolic Cosine of b x
Primitive of Exponential of a x by Hyperbolic Sine of b x

== Theorem ==
=== Exponential Form ===
=== Hyperbolic Form ===
== Also see ==
* Primitive of $e^{a x} \cosh b x$
Category:Primitive of Exponential of a x by Hyperbolic Sine of b x
Category:Primitives involving Exponential Function
Category:Primitives involving Hyperbolic Sine Function
Primitive of Exponential of a x by Hyperbolic Sine of b x/Exponential Form

== Theorem ==
:$\ds \int e^{a x} \sinh b x \rd x = \frac {e^{a x} } 2 \paren {\frac {e^{b x} } {a + b} - \frac {e^{-b x} } {a - b} } + C$
for $a^2 \ne b^2$.
== Proof ==
== Also see ==
* Primitive of $e^{a x} \cosh b x$
== Sources ==
* : Back endpapers: A Brief Table of Integrals: $137$.
Category:Primitive of Exponential of a x by Hyperbolic Sine of b x
Primitive of Exponential of a x by Hyperbolic Sine of b x/Hyperbolic Form

== Theorem ==
:$\ds \int e^{a x} \sinh b x \rd x = \frac {e^{a x} \paren {a \sinh b x - b \cosh b x} } {a^2 - b^2} + C$
for $a^2 \ne b^2$.
== Proof ==
== Also see ==
* Primitive of $e^{a x} \cosh b x$
Category:Primitive of Exponential of a x by Hyperbolic Sine of b x
Primitive of Exponential of a x by Logarithm of x

== Theorem ==
:$\ds \int e^{a x} \ln x \rd x = \frac {e^{a x} \ln x} a - \frac 1 a \int \frac {e^{a x} } x \rd x + C$
== 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 $\dfrac {e^{a x} } x$
== Sources ==
* : $\S 14$: Integrals involving $e^{a x}$: $14.522$
Category:Primitives involving Exponential Function
Category:Primitives involving Logarithm Function
Primitive of Exponential of a x by Power of Cosine of b x

== Theorem ==
:$\ds \int e^{a x} \cos^n b x \rd x = \frac {e^{a x} \cos^{n - 1} b x} {a^2 + n^2 b^2} \paren {a \cos b x + n b \sin b x} + \frac {n \paren {n - 1} b^2} {a^2 + n^2 b^2} \int e^{a x} \cos^{n - 2} b x \rd x + C$
== 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:
From Primitive of $e^{a x} \cos^{n - 1} b x \sin b x$:
:$\ds \int e^{a x} \cos^{n - 1} b x \sin b x \rd x = \frac {e^{a x} \cos^{n - 1} b x \paren {a \sin b x - b \cos b x} } {a^2 + n b^2} - \frac {\paren {n - 1} a b} {a^2 + n b^2} \paren {\int e^{a x} \cos^n b x \rd x - \int e^{a x} \cos^{n - 2} b x \rd x} + C$
Hence:
== Also see ==
* Primitive of $e^{a x} \sin^n b x$
== Sources ==
* : $\S 14$: Integrals involving $e^{a x}$: $14.524$
* : $\S 17$: Tables of Special Indefinite Integrals: $(25)$ Integrals Involving $e^{a x}$: $17.25.16.$
Category:Primitive of Exponential of a x by Power of Cosine of b x
Category:Primitives involving Exponential Function
Category:Primitives involving Cosine Function
Primitive of Exponential of a x by Power of Cosine of b x/Lemma 1

== Lemma for Primitive of $e^{a x} \cos b x$ ==
:$\ds \int e^{a x} \cos^{n - 1} b x \sin b x \rd x = \frac {e^{a x} \cos^{n - 1} b x \paren {a \sin b x - b \cos b x} } {a^2 + n b^2} - \frac {\paren {n - 1} a b} {a^2 + n b^2} \paren {\int e^{a x} \cos^n b x \rd x - \int e^{a x} \cos^{n - 2} b x \rd x} + C$
== 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:
and so rearranging:
:$\ds \int e^{a x} \cos^{n - 1} b x \sin b x \rd x = \frac {e^{a x} \cos^{n - 1} b x \paren {a \sin b x - b \cos b x} } {a^2 + n b^2} - \frac {\paren {n - 1} a b} {a^2 + n b^2} \paren {\int e^{a x} \cos^n b x \rd x - \int e^{a x} \cos^{n - 2} b x \rd x} + C$
Category:Primitive of Exponential of a x by Power of Cosine of b x
Primitive of Exponential of a x by Power of Cosine of b x/Lemma 2

== Lemma for Primitive of $e^{a x} \cos^n b x$ ==
:$\dfrac {a^2 + n b^2} a e^{a x} \cos^n b x + \dfrac {n b} a e^{a x} \cos^{n - 1} b x \paren {a \sin b x - b \cos b x} = e^{a x} \cos^{n - 1} b x \paren {a \cos b x + n b \sin b x}$
== Proof ==
Category:Primitive of Exponential of a x by Power of Cosine of b x
Primitive of Exponential of a x by Power of Sine of b x

== Theorem ==
:$\ds \int e^{a x} \sin^n b x \rd x = \frac {e^{a x} \sin^{n - 1} b x} {a^2 + n^2 b^2} \paren {a \sin b x - n b \cos b x} + \frac {n \paren {n - 1} b^2} {a^2 + n^2 b^2} \int e^{a x} \sin^{n - 2} b x \rd x + C$
== 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:
From Primitive of $e^{a x} \sin^{n - 1} b x \cos b x$: Lemma 1:
:$\ds \int e^{a x} \sin^{n - 1} b x \cos b x \rd x = \frac {e^{a x} \sin^{n - 1} b x \paren {a \cos b x + b \sin b x} } {a^2 + n b^2} + \frac {\paren {n - 1} a b} {a^2 + n b^2} \paren {\int e^{a x} \sin^n b x \rd x - \int e^{a x} \sin^{n - 2} b x \rd x} + C$
Hence:
== Also see ==
* Primitive of $e^{a x} \cos^n b x$
== Sources ==
* : $\S 14$: Integrals involving $e^{a x}$: $14.523$
Category:Primitives involving Exponential Function
Category:Primitives involving Sine Function
Category:Primitive of Exponential of a x by Power of Sine of b x
Primitive of Exponential of a x by Power of Sine of b x/Lemma 1

== Lemma for Primitive of $e^{a x} \sin^n b x \cos b x$ ==
:$\ds \int e^{a x} \sin^{n - 1} b x \cos b x \rd x = \frac {e^{a x} \sin^{n - 1} b x \paren {a \cos b x + b \sin b x} } {a^2 + n b^2} + \frac {\paren {n - 1} a b} {a^2 + n b^2} \paren {\int e^{a x} \sin^n b x \rd x - \int e^{a x} \sin^{n - 2} b x \rd x} + C$
== 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:
and so rearranging:
:$\ds \int e^{a x} \sin^{n - 1} b x \cos b x \rd x = \frac {e^{a x} \sin^{n - 1} b x \paren {a \cos b x + b \sin b x} } {a^2 + n b^2} + \frac {\paren {n - 1} a b} {a^2 + n b^2} \paren {\int e^{a x} \sin^n b x \rd x - \int e^{a x} \sin^{n - 2} b x \rd x} + C$
Category:Primitive of Exponential of a x by Power of Sine of b x
Primitive of Exponential of a x by Power of Sine of b x/Lemma 2

== Lemma for Primitive of $e^{a x} \sin^n b x$ ==
:$\dfrac {a^2 + n b^2} a e^{a x} \sin^n b x - \dfrac {n b} a e^{a x} \sin^{n - 1} b x \left({a \cos b x + b \sin b x}\right) = e^{a x} \sin^{n - 1} b x \left({a \sin b x - n b \cos b x}\right)$
== Proof ==
Category:Primitive of Exponential of a x by Power of Sine of b x
Primitive of Exponential of a x by Sine of b x

== Theorem ==
:$\ds \int e^{a x} \sin b x \rd x = \frac {e^{a x} \paren {a \sin b x - b \cos b x} } {a^2 + b^2} + C$
== Proof 1 ==
== Proof 2 ==
== Proof 3 ==
== Proof 4 ==
== Also see ==
* Primitive of $e^{a x} \cos b x$
== Sources ==
* : $\S 14$: Integrals involving $e^{a x}$: $14.518$
* : Back endpapers: A Brief Table of Integrals: $107$.
* : Appendix $7$: Integrals
* : Appendix $8$: Integrals
Category:Primitive of Exponential of a x by Sine of b x
Category:Primitives involving Exponential Function
Category:Primitives involving Sine Function
Primitive of Exponential of a x by Sine of b x/Lemma

== Lemma for Primitive of $e^{a x} \sin b x$ ==
:$\ds \int e^{a x} \sin b x \rd x = \frac {e^{a x} \sin b x} a - \frac b a \int e^{a x} \cos b x \rd x$
== 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:
Category:Primitive of Exponential of a x by Sine of b x
Primitive of Exponential of a x by Sine of b x/Proof 4

== Theorem == 
== Proof ==
Let $a, b, x \in \R$ be real numbers.
Suppose $a \ne 0 \ne b$.
Let $f_1$ and $f_2$ be the real functions defined as:
Let $\map \CC \R$ be the space of continuous real-valued functions.
Let $\struct {\map {\CC^1} \R, +, \, \cdot \,}_\R$ be the vector space of continuously differentiable real-valued functions.
Let $S = \span \set {f_1, f_2} \subset \map {\CC^1} \R$ be a vector space.
Let $D : S \to S$ be the derivative $x$.
From Differentiation of Exponential of a x by Cosine of b x and Exponential of a x by Sine of b x wrt x as Nonsingular Matrix, $D$ is expressible as:
:$\mathbf D = \begin {bmatrix} a & b \\ -b & a \end {bmatrix}$
and is nonsingular.
By Inverse of Matrix is Scalar Product of Adjugate by Reciprocal of Determinant:
:$\mathbf D^{-1} = \dfrac 1 {a^2 + b^2} \begin {bmatrix} a & -b \\ b & a \end{bmatrix}$
Then:
Application of $\mathbf D$ on both sides on the left and writing out explicitly in terms of $f_1$ and $f_2$ yields:
:$f_2 = \ds \dfrac \d {\d x} \frac {-b f_1 + a f_2} {a^2 + b^2}$
Integrate $x$:
:$\ds \int f_2 \rd x = \frac {-b f_1 + a f_2} {a^2 + b^2} + C$
where $C$ is an arbitrary constant.
Substitute definitions of $f_1$ and $f_2$ to get the desired result.
== Sources ==
* : Chapter $\S 2.1$: Continuous and linear maps. Linear transformations
Category:Primitive of Exponential of a x by Sine of b x
Primitive of Exponential of a x over Power of x

== Theorem ==
:$\ds \int \frac {e^{a x} \rd x} {x^n} = \frac {-e^{a x} } {\paren {n - 1} x^{n - 1} } + \frac a {n - 1} \int \frac {e^{a x} \rd x} {x^{n - 1} } + C$
where $n \ne 1$.
== 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:
Then:
== Also see ==
* Primitive of $\dfrac {e^{a x} } x$ for the case where $n = 1$
* Primitive of $x^n e^{a x}$
== Sources ==
* : $\S 14$: Integrals involving $e^{a x}$: $14.514$
Category:Primitives involving Exponential Function
Primitive of Exponential of a x over x

== Theorem ==
For $x > 0$:
== Proof ==
The validity of $(1)$ follows from absolute convergence of the power series expansion.
== Also see ==
* Primitive of $\dfrac {e^x} x$ has no Solution in Elementary Functions
== Sources ==
* : $\S 14$: Integrals involving $e^{a x}$: $14.513$
Category:Primitives involving Exponential Function
Primitive of Fourth Power of Cosine Function

== Theorem ==
:$\ds \int \cos^4 x \rd x = \frac {3 x} 8 + \frac {\sin 2 x} 4 + \frac {\sin 4 x} {32} + C$
== Proof ==
From Primitive of $\cos^4 a x$:
The result follows by setting $a = 1$.
== Sources ==
* : $\text {II}$. Calculus: Exercises $\text {XIV}$: $10$.
Category:Primitives involving Cosine Function
Primitive of Fourth Power of Cosine of a x

== Theorem ==
:$\ds \int \cos^4 a x \rd x = \frac {3 x} 8 + \frac {\sin 2 a x} {4 a} + \frac {\sin 4 a x} {32 a} + C$
== Proof ==
== Also see ==
* Primitive of $\sin^4 a x$
== Sources ==
* : $\S 14$: Integrals involving $\cos a x$: $14.380$
Category:Primitives involving Cosine Function
Primitive of Fourth Power of Sine of a x

== Theorem ==
:$\ds \int \sin^4 a x \rd x = \frac {3 x} 8 - \frac {\sin 2 a x} {4 a} + \frac {\sin 4 a x} {32 a} + C$
== Proof ==
== Also see ==
* Primitive of $\cos^4 a x$
== Sources ==
* : $\S 14$: Integrals involving $\sin a x$: $14.350$
Category:Primitives involving Sine Function
Primitive of Function of Arccosecant

== Theorem ==
:$\ds \int \map F {\arccsc \frac x a} \rd x = -a \int \map F u \size {\csc u} \cot u \rd u$
where $u = \arccsc \dfrac x a$.
== Proof ==
First note that:
Then:
== Also see ==
* Primitive of Function of Arcsine
* Primitive of Function of Arccosine
* Primitive of Function of Arctangent
* Primitive of Function of Arccotangent
* Primitive of Function of Arcsecant
Category:Integral Substitutions
Category:Arccosecant Function
Category:Primitives involving Inverse Cosecant Function
Primitive of Function of Arccosine

== Theorem ==
:$\ds \int \map F {\arccos \frac x a} \rd x = -a \int \map F u \sin u \rd u$
where $u = \arccos \dfrac x a$.
== Proof ==
First note that:
Then:
== Also see ==
* Primitive of Function of Arcsine
* Primitive of Function of Arctangent
* Primitive of Function of Arccotangent
* Primitive of Function of Arcsecant
* Primitive of Function of Arccosecant
Category:Integral Substitutions
Category:Arccosine Function
Category:Primitives involving Inverse Cosine Function
Primitive of Function of Arccotangent

== Theorem ==
:$\ds \int \map F {\arccot \frac x a} \rd x = -a \int \map F u \csc^2 u \rd u$
where $u = \arccot \dfrac x a$.
== Proof ==
First note that:
Then:
== Also see ==
* Primitive of Function of Arcsine
* Primitive of Function of Arccosine
* Primitive of Function of Arctangent
* Primitive of Function of Arcsecant
* Primitive of Function of Arccosecant
Category:Integral Substitutions
Category:Arccotangent Function
Category:Primitives involving Inverse Cotangent Function
Primitive of Function of Arcsecant

== Theorem ==
:$\ds \int \map F {\arcsec \frac x a} \rd x = a \int \map F u \sec u \tan u \rd u$
where $u = \arcsec \dfrac x a$.
== Proof ==
First note that:
Then:
== Also see ==
* Primitive of Function of Arcsine
* Primitive of Function of Arccosine
* Primitive of Function of Arctangent
* Primitive of Function of Arccotangent
* Primitive of Function of Arccosecant
Category:Integral Substitutions
Category:Arcsecant Function
Category:Primitives involving Inverse Secant Function
Primitive of Function of Arcsine

== Theorem ==
:$\ds \int \map F {\arcsin \frac x a} \rd x = a \int \map F u \cos u \rd u$
where $u = \arcsin \dfrac x a$.
== Proof ==
First note that:
Then:
== Also see ==
* Primitive of Function of Arccosine
* Primitive of Function of Arctangent
* Primitive of Function of Arccotangent
* Primitive of Function of Arcsecant
* Primitive of Function of Arccosecant
== Sources ==
* : $\S 14$: Important Transformations: $14.57$
* : $\S 16$: Indefinite Integrals: Important Transformations: $16.57.$
Category:Integral Substitutions
Category:Arcsine Function
Category:Primitives involving Inverse Sine Function
Primitive of Function of Arctangent

== Theorem ==
:$\ds \int \map F {\arctan \frac x a} \rd x = a \int \map F u \sec^2 u \rd u$
where $u = \arctan \dfrac x a$.
== Proof ==
First note that:
Then:
== Also see ==
* Primitive of Function of Arcsine
* Primitive of Function of Arccosine
* Primitive of Function of Arccotangent
* Primitive of Function of Arcsecant
* Primitive of Function of Arccosecant
Category:Integral Substitutions
Category:Arctangent Function
Category:Primitives involving Inverse Tangent Function
Primitive of Function of Constant Multiple

== Theorem ==
Let $f$ be a real function which is integrable.
Let $c$ be a constant.
Then:
:$\ds \int \map f {c x} \rd x = \frac 1 c \int \map f u \d u$
where $u = c x$.
== Proof ==
Let $u = c x$.
By :
:$\dfrac {\d u} {\d x} = c$
Thus:
== Sources ==
* : $\S 14$: General Rules of Integration: $14.5$
* : $\S 16$: Indefinite Integrals: General Rules of Integration: $16.5.$
Category:Primitives
Primitive of Function of Exponential Function

== Theorem ==
:$\ds \int \map F {e^{a x} } \rd x = \frac 1 a \int \frac {\map F u} u \rd u$
where $u = e^{a x}$.
== Proof ==
== Sources ==
* : $\S 14$: Important Transformations: $14.55$
* : $\S 16$: Indefinite Integrals: Important Transformations: $16.55.$
Category:Integral Substitutions
Primitive of Function of Natural Logarithm

== Theorem ==
:$\ds \int \map F {\ln x} \rd x = \int \map F u e^u \rd u$
where $u = \ln x$.
== Proof ==
== Sources ==
* : $\S 14$: Important Transformations: $14.56$
* : $\S 16$: Indefinite Integrals: Important Transformations: $16.56.$
Category:Integral Substitutions
Primitive of Function of Nth Root of a x + b

== Theorem ==
:$\ds \int \map F {\sqrt [n] {a x + b} } \rd x = \frac n a \int u^{n - 1} \map F u \rd u$
where $u = \sqrt [n] {a x + b}$.
== Proof ==
== Sources ==
* : $\S 14$: Important Transformations: $14.51$
* : $\S 16$: Indefinite Integrals: Important Transformations: $16.51.$
Category:Integral Substitutions
Primitive of Function of Root of a squared minus x squared

== Theorem ==
:$\ds \int \map F {\sqrt {a^2 - x^2} } \rd x = a \int \map F {a \cos u} \cos u \rd u$
where $x = a \sin u$.
== Proof ==
First note that:
Then:
== Sources ==
* : $\S 14$: Important Transformations: $14.52$
* : $\S 16$: Indefinite Integrals: Important Transformations: $16.52.$
Category:Integral Substitutions
Primitive of Function of Root of a squared plus x squared

== Theorem ==
:$\ds \int \map F {\sqrt {a^2 + x^2} } \rd x = a \int \map F {a \sec u} \sec^2 u \rd u$
where $x = a \tan u$.
== Proof ==
First note that:
Then:
== Sources ==
* : $\S 14$: Important Transformations: $14.53$
* : $\S 16$: Indefinite Integrals: Important Transformations: $16.53.$
Category:Integral Substitutions
Primitive of Function of Root of a x + b

== Theorem ==
:$\ds \int \map F {\sqrt {a x + b} } \rd x = \frac 2 a \int u \map F u \rd u$
where $u = \sqrt {a x + b}$.
== Proof ==
== Sources ==
* : $\S 14$: Important Transformations: $14.50$
* : $\S 16$: Indefinite Integrals: Important Transformations: $16.50.$
Category:Integral Substitutions
Category:Primitives involving Root of a x + b
Primitive of Function of Root of x squared minus a squared

== Theorem ==
:$\ds \int \map F {\sqrt {x^2 - a^2} } \rd x = a \int \map F {a \tan u} \sec u \tan u \rd u$
where $x = a \sec u$.
== Proof ==
First note that:
Then:
== Sources ==
* : $\S 14$: Important Transformations: $14.54$
* : $\S 16$: Indefinite Integrals: Important Transformations: $16.54.$
Category:Integral Substitutions
Primitive of Function of a x + b

== Theorem ==
:$\ds \int \map F {a x + b} \rd x = \frac 1 a \int \map F u \rd u$
where $u = a x + b$.
== Proof ==
== Sources ==
* : Chapter $\text {III}$: Integration: Three rules for integration: $\text {III}$
* : $\text {II}$. Calculus: Integration
* : $\S 14$: Important Transformations: $14.49$
* : $\S 16$: Indefinite Integrals: Important Transformations: $16.49.$
Category:Integral Substitutions
Category:Primitives involving a x + b