nyuuzyou/wikis huggingface dataset:
Primitive of Sine of x over x has no Solution in Elementary Functions

== Theorem ==
The primitive:
:$\ds \int \frac {\sin x \rd x} x$
cannot be expressed in terms of a finite number of elementary functions.
== Proof ==
== Also see ==
* Primitive of $\dfrac {\sin a x} x$
== Historical Note ==
== Sources ==
* : Chapter $\text {A}.29$: Liouville ()
Category:Primitives involving Sine Function
Primitive of Sine of x over x has no Solution in Elementary Functions/Historical Note

== Historical Note on Primitive of $\dfrac {\sin x} x$ has no Solution in Elementary Functions ==
The proof that $\ds \int \dfrac {\sin x \rd x} x$ 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 Sine x by Logarithm of Sine x

== Theorem ==
:$\ds \int \sin x \map \ln {\sin x} \rd x = \cos x \paren {1 - \map \ln {\sin x} } + \ln \size {\tan \frac x 2} + C$
== Proof ==
We have: 
We also have, by Primitive of Sine Function:
:$\ds \int \sin x \rd x = -\cos x + C$
So: 
Category:Primitives involving Sine Function
Category:Primitives involving Logarithm Function
Primitive of Square of Arccosine of x over a

== Theorem ==
:$\ds \int \paren {\arccos \frac x a}^2 \rd x = x \paren {\arccos \frac x a}^2 - 2 x - 2 \sqrt {a^2 - x^2} \arccos \frac x a + C$
== Proof ==
Let:
Then:
== Also see ==
* Primitive of $\paren {\arcsin \dfrac x a}^2$
== Sources ==
* : $\S 14$: Integrals involving Inverse Trigonometric Functions: $14.482$
Category:Primitives involving Inverse Cosine Function
Primitive of Square of Arcsine of x over a

== Theorem ==
:$\ds \int \paren {\arcsin \frac x a}^2 \rd x = x \paren {\arcsin \frac x a}^2 - 2 x + 2 \sqrt{a^2 - x^2} \arcsin \frac x a + C$
== Proof ==
Let:
Then:
== Also see ==
* Primitive of $\paren {\arccos \dfrac x a}^2$
== Sources ==
* : $\S 14$: Integrals involving Inverse Trigonometric Functions: $14.476$
Category:Primitives involving Inverse Sine Function
Primitive of Square of Cosecant Function

== Theorem ==
:$\ds \int \csc^2 x \rd x = -\cot x + C$
where $C$ is an arbitrary constant.
== Proof ==
From Derivative of Cotangent Function:
:$\dfrac \d {\d x} \cot x = -\csc^2 x$
The result follows from the definition of primitive.
== Also see ==
* Primitive of Square of Secant Function
== Sources ==
* : Chapter $\text {II}$: Integration of Elementary Functions: $\S 7$. Standard Integrals: $7$.
* : Chapter $\text {III}$: Integration: Integration
* : Chapter $10$: Integration: $10.4$. Standard integrals: Other Standard Results: $\text {(xiv)}$
* : $\S 14$: General Rules of Integration: $14.18$
* : Chapter $25$: Fundamental Integration Formulas: $15$.
* : Chapter $5$. Integrals: Integrals of Special Functions: $10$
* : Appendix: Table $2$: Integrals
* : Appendix: Table $2$: Integrals
* : $\S 16$: Indefinite Integrals: General Rules of Integration: $16.18.$
Category:Primitives involving Cosecant Function
Category:Cotangent Function
Primitive of Square of Cosecant of a x

== Theorem ==
:$\ds \int \csc^2 a x \rd x = -\frac {\cot a x} a + C$
== Proof ==
== Also see ==
* Primitive of $\sin^2 a x$
* Primitive of $\cos^2 a x$
* Primitive of $\tan^2 a x$
* Primitive of $\cot^2 a x$
* Primitive of $\sec^2 a x$
== Sources ==
* : $\S 14$: Integrals involving $\csc a x$: $14.462$
* : Back endpapers: A Brief Table of Integrals: $91$.
* : $\S 17$: Tables of Special Indefinite Integrals: $(23)$ Integrals Involving $\csc a x$: $17.23.2.$
Category:Primitives involving Cosecant Function
Primitive of Square of Cosecant of a x over Cotangent of a x

== Theorem ==
:$\ds \int \frac {\csc^2 a x \rd x} {\cot a x} = \frac {-\ln \size {\cot a x} } a + C$
== Proof ==
== Also see ==
* Primitive of $\dfrac {\sec^2 a x} {\tan a x}$
== Sources ==
* : $\S 14$: Integrals involving $\cot a x$: $14.444$
Category:Primitives involving Cotangent Function
Category:Primitives involving Cosecant Function
Primitive of Square of Cosine Function

== Theorem ==
:$\ds \int \cos^2 x \rd x = \frac x 2 + \frac {\sin 2 x} 4 + C$
where $C$ is an arbitrary constant.
=== Corollary ===
== Proof 1 ==
== Proof 2 ==
== Proof 3 ==
== Also presented as ==
Some sources present this as:
:$\ds \int \cos^2 x \rd x = \frac 1 2 \paren {x + \frac {\sin 2 x} 2} + C$
== Sources ==
* : Chapter $10$: Integration: $10.4$. Standard integrals: Other Standard Results: $\text {(xxii)}$
* : $\S 14$: General Rules of Integration: $14.22$
* : Appendix: Table $2$: Integrals
* : Appendix: Table $2$: Integrals
* : $\S 16$: Indefinite Integrals: General Rules of Integration: $16.22.$
* : Appendix $7$: Integrals
* : Appendix $8$: Integrals
Category:Primitive of Square of Cosine Function
Category:Primitives involving Cosine Function
Primitive of Square of Cosine Function/Corollary

== Corollary to Primitive of Square of Cosine Function ==
:$\ds \int \cos^2 x \rd x = \frac {x + \sin x \cos x} 2 + C$
where $C$ is an arbitrary constant.
== Proof ==
== Sources ==
* : $\S 14$: General Rules of Integration: $14.22$
* : $6$. Integral Calculus: Appendix: Table of Fundamental Standard Integrals
* : $\S 16$: Indefinite Integrals: General Rules of Integration: $16.22.$
Category:Primitive of Square of Cosine Function
Primitive of Square of Cosine of a x

== Theorem ==
:$\ds \int \cos^2 a x \rd x = \frac x 2 + \frac {\sin 2 a x} {4 a} + C$
== Proof ==
== Also see ==
* Primitive of $\sin^2 a x$
* Primitive of $\tan^2 a x$
* Primitive of $\cot^2 a x$
* Primitive of $\sec^2 a x$
* Primitive of $\csc^2 a x$
== Sources ==
* : $\S 14$: Integrals involving $\cos a x$: $14.377$
* : Front endpapers: A Brief Table of Integrals: $59$.
Category:Primitives involving Cosine Function
Primitive of Square of Cosine of a x over Sine of a x

== Theorem ==
:$\ds \int \frac {\cos^2 a x \rd x} {\sin a x} = \frac {\cos a x} a + \frac 1 a \ln \size {\tan \frac {a x} 2} + C$
== Proof ==
== Also see ==
* Primitive of $\dfrac {\sin^2 a x} {\cos a x}$
== Sources ==
* : $\S 14$: Integrals involving $\sin a x$ and $\cos a x$: $14.409$
Category:Primitives involving Sine Function and Cosine Function
Primitive of Square of Cotangent Function

== Theorem ==
:$\ds \int \cot^2 x \rd x = -\cot x - x + C$
where $C$ is an arbitrary constant.
== Proof ==
== Sources ==
* : $\S 14$: General Rules of Integration: $14.20$
* : $6$. Integral Calculus: Appendix: Table of Fundamental Standard Integrals
* : $\S 16$: Indefinite Integrals: General Rules of Integration: $16.20.$
Category:Primitives involving Cotangent Function
Primitive of Square of Cotangent of a x

== Theorem ==
:$\ds \int \cot^2 a x \rd x = \frac {-\cot a x} a - x + C$
== Proof ==
== Also see ==
* Primitive of $\sin^2 a x$
* Primitive of $\cos^2 a x$
* Primitive of $\tan^2 a x$
* Primitive of $\sec^2 a x$
* Primitive of $\csc^2 a x$
== Sources ==
* : $\S 14$: Integrals involving $\cot a x$: $14.441$
* : Back endpapers: A Brief Table of Integrals: $85$.
* : $\S 17$: Tables of Special Indefinite Integrals: $(21)$ Integrals Involving $\cot a x$: $17.21.2.$
Category:Primitives involving Cotangent Function
Primitive of Square of Hyperbolic Cosecant Function

== Theorem ==
:$\ds \int \csch^2 x \rd x = -\coth x + C$
where $C$ is an arbitrary constant.
== Proof ==
From Derivative of Hyperbolic Cotangent:
:$\map {\dfrac \d {\d x} } {\coth x} = -\csch^2 x$
The result follows from the definition of primitive.
== Sources ==
* : Chapter $\text {II}$: Integration of Elementary Functions: $\S 7$. Standard Integrals: $11$.
* : Chapter $10$: Integration: $10.4$. Standard integrals: Other Standard Results: $\text {(xxx)}$
* : $\S 14$: General Rules of Integration: $14.32$
* : Chapter $5$. Integrals: Integrals of Special Functions: $22$
* : $\S 16$: Indefinite Integrals: General Rules of Integration: $16.32.$
Category:Primitives of Hyperbolic Functions
Category:Hyperbolic Cosecant Function
Category:Hyperbolic Cotangent Function
Primitive of Square of Hyperbolic Cosecant of a x

== Theorem ==
:$\ds \int \csch^2 a x \rd x = \frac {-\coth a x} a + C$
== Proof ==
== Also see ==
* Primitive of $\sinh^2 a x$
* Primitive of $\cosh^2 a x$
* Primitive of $\tanh^2 a x$
* Primitive of $\coth^2 a x$
* Primitive of $\sech^2 a x$
== Sources ==
* : $\S 14$: Integrals involving $\csch a x$: $14.637$
* : Back endpapers: A Brief Table of Integrals: $132$.
* : $\S 17$: Tables of Special Indefinite Integrals: $(32)$ Integrals Involving $\csch a x$: $17.33.2.$
Category:Primitives involving Hyperbolic Cosecant Function
Primitive of Square of Hyperbolic Cosecant of a x over Hyperbolic Cotangent of a x

== Theorem ==
:$\ds \int \frac {\csch^2 a x \rd x} {\coth a x} = \frac {-1} a \ln \size {\coth a x} + C$
== Proof ==
== Also see ==
* Primitive of $\dfrac {\sech^2 a x \rd x} {\tanh a x}$
== Sources ==
* : $\S 14$: Integrals involving $\coth a x$: $14.619$
Category:Primitives involving Hyperbolic Cotangent Function
Category:Primitives involving Hyperbolic Cosecant Function
Primitive of Square of Hyperbolic Cosine Function

== Theorem ==
:$\ds \int \cosh^2 x \rd x = \frac {\sinh 2 x} 4 + \frac x 2 + C$
where $C$ is an arbitrary constant.
=== Corollary ===
== Proof ==
== Sources ==
* : $\S 14$: General Rules of Integration: $14.36$
* : $\S 16$: Indefinite Integrals: General Rules of Integration: $16.36.$
Category:Primitive of Square of Hyperbolic Cosine Function
Category:Primitives of Hyperbolic Functions
Category:Hyperbolic Cosine Function
Primitive of Square of Hyperbolic Cosine Function/Corollary

== Corollary to Primitive of Square of Hyperbolic Cosine Function ==
:$\ds \int \cosh^2 x \rd x = \frac {\sinh x \cosh x + x} 2 + C$
where $C$ is an arbitrary constant.
== Proof ==
== Sources ==
* : $\S 14$: General Rules of Integration: $14.36$
* : $\S 16$: Indefinite Integrals: General Rules of Integration: $16.36.$
Category:Primitive of Square of Hyperbolic Cosine Function
Primitive of Square of Hyperbolic Cosine of a x

== Theorem ==
:$\ds \int \cosh^2 a x \rd x = \frac {\sinh a x \cosh a x} {2 a} + \frac x 2 + C$
=== Corollary ===
== Proof ==
== Also see ==
* Primitive of $\sinh^2 a x$
* Primitive of $\tanh^2 a x$
* Primitive of $\coth^2 a x$
* Primitive of $\sech^2 a x$
* Primitive of $\csch^2 a x$
== Sources ==
* : $\S 14$: Integrals involving $\cosh a x$: $14.569$
Category:Primitive of Square of Hyperbolic Cosine of a x
Category:Primitives involving Hyperbolic Cosine Function
Primitive of Square of Hyperbolic Cosine of a x/Corollary

== Theorem ==
:$\ds \int \cosh^2 a x \rd x = \frac {\sinh 2 a x} {4 a} + \frac x 2 + C$
== Proof ==
== Also see ==
* Primitive of $\sinh^2 a x$
* Primitive of $\tanh^2 a x$
* Primitive of $\coth^2 a x$
* Primitive of $\sech^2 a x$
* Primitive of $\csch^2 a x$
== Sources ==
* : Back endpapers: A Brief Table of Integrals: $116$.
Category:Primitive of Square of Hyperbolic Cosine of a x
Primitive of Square of Hyperbolic Cosine of a x over Hyperbolic Sine of a x

== Theorem ==
:$\ds \int \frac {\cosh^2 a x \rd x} {\sinh a x} = \frac {\cosh a x} a + \frac 1 a \ln \size {\tanh \frac {a x} 2} + C$
== Proof ==
== Sources ==
* : $\S 14$: Integrals involving $\sinh a x $ and $\cosh a x$: $14.600$
* : $\S 17$: Tables of Special Indefinite Integrals: $(29)$ Integrals Involving $\sinh a x $ and $\cosh a x$: $17.29.7.$
Category:Primitives involving Hyperbolic Sine Function
Category:Primitives involving Hyperbolic Cosine Function
Primitive of Square of Hyperbolic Cotangent Function

== Theorem ==
:$\ds \int \coth^2 x \rd x = x - \coth x + C$
where $C$ is an arbitrary constant.
== Proof ==
== Sources ==
* : $\S 14$: General Rules of Integration: $14.34$
* : $\S 16$: Indefinite Integrals: General Rules of Integration: $16.34.$
Category:Primitives of Hyperbolic Functions
Category:Hyperbolic Cotangent Function
Primitive of Square of Hyperbolic Cotangent of a x

== Theorem ==
:$\ds \int \coth^2 a x \rd x = x - \frac {\coth a x} a + C$
== Proof ==
== Also see ==
* Primitive of $\sinh^2 a x$
* Primitive of $\cosh^2 a x$
* Primitive of $\tanh^2 a x$
* Primitive of $\sech^2 a x$
* Primitive of $\csch^2 a x$
== Sources ==
* : $\S 14$: Integrals involving $\coth a x$: $14.616$
* : Back endpapers: A Brief Table of Integrals: $126$.
* : $\S 17$: Tables of Special Indefinite Integrals: $(31)$ Integrals Involving $\coth a x$: $17.31.2.$
Category:Primitives involving Hyperbolic Cotangent Function
Primitive of Square of Hyperbolic Secant Function

== Theorem ==
:$\ds \int \sech^2 x \rd x = \tanh x + C$
where $C$ is an arbitrary constant.
== Proof ==
From Derivative of Hyperbolic Tangent:
:$\map {\dfrac \d {\d x} } {\tanh x} = \sech^2 x$
The result follows from the definition of primitive.
== Sources ==
* : Chapter $\text {II}$: Integration of Elementary Functions: $\S 7$. Standard Integrals: $10$.
* : Chapter $10$: Integration: $10.4$. Standard integrals: Other Standard Results: $\text {(xxix)}$
* : $\S 14$: Indefinite Integrals: General Rules of Integration: $14.31$
* : Chapter $5$. Integrals: Integrals of Special Functions: $21$
* : $\S 16$: Indefinite Integrals: General Rules of Integration: $16.31.$
Category:Primitives of Hyperbolic Functions
Category:Hyperbolic Secant Function
Category:Hyperbolic Tangent Function
Primitive of Square of Hyperbolic Secant of a x

== Theorem ==
:$\ds \int \sech^2 a x \rd x = \frac {\tanh a x} a + C$
== Proof ==
== Also see ==
* Primitive of $\sinh^2 a x$
* Primitive of $\cosh^2 a x$
* Primitive of $\tanh^2 a x$
* Primitive of $\coth^2 a x$
* Primitive of $\csch^2 a x$
== Sources ==
* : $\S 14$: Integrals involving $\sech a x$: $14.627$
* : Back endpapers: A Brief Table of Integrals: $131$.
* : $\S 17$: Tables of Special Indefinite Integrals: $(33)$ Integrals Involving $\sech a x$: $17.32.2.$
Category:Primitives involving Hyperbolic Secant Function
Primitive of Square of Hyperbolic Secant of a x over Hyperbolic Tangent of a x

== Theorem ==
:$\ds \int \frac {\sech^2 a x \rd x} {\tanh a x} = \frac 1 a \ln \size {\tanh a x} + C$
== Proof ==
== Also see ==
* Primitive of $\dfrac {\csch^2 a x \rd x} {\coth a x}$
== Sources ==
* : $\S 14$: Integrals involving $\tanh a x$: $14.608$
Category:Primitives involving Hyperbolic Tangent Function
Category:Primitives involving Hyperbolic Secant Function
Primitive of Square of Hyperbolic Sine Function

== Theorem ==
:$\ds \int \sinh^2 x \rd x = \frac {\sinh 2 x} 4 - \frac x 2 + C$
where $C$ is an arbitrary constant.
=== Corollary ===
== Proof ==
== Sources ==
* : $\S 14$: General Rules of Integration: $14.35$
* : $\S 16$: Indefinite Integrals: General Rules of Integration: $16.35.$
Category:Primitive of Square of Hyperbolic Sine Function
Category:Primitives of Hyperbolic Functions
Category:Hyperbolic Sine Function
Primitive of Square of Hyperbolic Sine Function/Corollary

== Corollary to Primitive of Square of Hyperbolic Sine Function ==
:$\ds \int \sinh^2 x \rd x = \frac {\sinh x \cosh x - x} 2 + C$
where $C$ is an arbitrary constant.
== Proof ==
== Sources ==
* : $\S 14$: General Rules of Integration: $14.35$
* : $\S 16$: Indefinite Integrals: General Rules of Integration: $16.35.$
Category:Primitive of Square of Hyperbolic Sine Function
Primitive of Square of Hyperbolic Sine of a x

== Theorem ==
:$\ds \int \sinh^2 a x \rd x = \dfrac {\sinh a x \cosh a x} {2 a} - \frac x 2 + C$
=== Corollary ===
== Proof ==
== Also see ==
* Primitive of $\cosh^2 a x$
* Primitive of $\tanh^2 a x$
* Primitive of $\coth^2 a x$
* Primitive of $\sech^2 a x$
* Primitive of $\csch^2 a x$
== Sources ==
* : $\S 14$: Integrals involving $\sinh a x$: $14.547$
Category:Primitive of Square of Hyperbolic Sine of a x
Category:Primitives involving Hyperbolic Sine Function
Primitive of Square of Hyperbolic Sine of a x/Corollary

== Theorem ==
:$\ds \int \sinh^2 a x \rd x = \frac {\sinh 2 a x} {4 a} - \frac x 2 + C$
== Proof ==
== Also see ==
* Primitive of $\cosh^2 a x$
* Primitive of $\tanh^2 a x$
* Primitive of $\coth^2 a x$
* Primitive of $\sech^2 a x$
* Primitive of $\csch^2 a x$
== Sources ==
* : Back endpapers: A Brief Table of Integrals: $115$.
Category:Primitive of Square of Hyperbolic Sine of a x
Primitive of Square of Hyperbolic Sine of a x by Square of Hyperbolic Cosine of a x

== Theorem ==
:$\ds \int \sinh^2 a x \cosh^2 a x \rd x = \frac {\sinh 4 a x} {32 a} - \frac x 8 + C$
== Proof ==
== Sources ==
* : $\S 14$: Integrals involving $\sinh a x $ and $\cosh a x$: $14.594$
* : $\S 17$: Tables of Special Indefinite Integrals: $(29)$ Integrals Involving $\sinh a x $ and $\cosh a x$: $17.29.3.$
Category:Primitives involving Hyperbolic Sine Function
Category:Primitives involving Hyperbolic Cosine Function
Primitive of Square of Hyperbolic Sine of a x over Hyperbolic Cosine of a x

== Theorem ==
:$\ds \int \frac {\sinh^2 a x \rd x} {\cosh a x} = \frac {\sinh a x} a - \frac 1 a \map \arctan {\sinh a x} + C$
== Proof ==
== Sources ==
* : $\S 14$: Integrals involving $\sinh a x $ and $\cosh a x$: $14.599$
* : $\S 17$: Tables of Special Indefinite Integrals: $(29)$ Integrals Involving $\sinh a x $ and $\cosh a x$: $17.29.6.$
Category:Primitives involving Hyperbolic Sine Function
Category:Primitives involving Hyperbolic Cosine Function
Primitive of Square of Hyperbolic Tangent Function

== Theorem ==
:$\ds \int \tanh^2 x \rd x = x - \tanh x + C$
where $C$ is an arbitrary constant.
== Proof ==
== Sources ==
* : $\S 14$: General Rules of Integration: $14.33$
* : $\S 16$: Indefinite Integrals: General Rules of Integration: $16.33.$
Category:Primitives of Hyperbolic Functions
Category:Hyperbolic Tangent Function
Primitive of Square of Hyperbolic Tangent of a x

== Theorem ==
:$\ds \int \tanh^2 a x \rd x = x - \frac {\tanh a x} a + C$
== Proof ==
== Also see ==
* Primitive of $\sinh^2 a x$
* Primitive of $\cosh^2 a x$
* Primitive of $\coth^2 a x$
* Primitive of $\sech^2 a x$
* Primitive of $\csch^2 a x$
== Sources ==
* : $\S 14$: Integrals involving $\tanh a x$: $14.605$
* : Back endpapers: A Brief Table of Integrals: $125$.
* : $\S 17$: Tables of Special Indefinite Integrals: $(30)$ Integrals Involving $\tanh a x$: $17.30.2.$
Category:Primitives involving Hyperbolic Tangent Function
Primitive of Square of Logarithm of x

== Theorem ==
:$\ds \int \ln^2 x \rd x = x \ln^2 x - 2 x \ln x + 2 x + C$
== 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:
== Sources ==
* : $\S 14$: Integrals involving $\ln x$: $14.530$
Category:Primitives involving Logarithm Function
Primitive of Square of Secant Function

== Theorem ==
:$\ds \int \sec^2 x \rd x = \tan x + C$
where $C$ is an arbitrary constant.
== Proof ==
From Derivative of Tangent Function:
:$\map {\dfrac \d {\d x} } {\tan x} = \sec^2 x$
The result follows from the definition of primitive.
== Also see ==
* Primitive of Square of Cosecant Function
== Sources ==
* : Chapter $\text {II}$: Integration of Elementary Functions: $\S 7$. Standard Integrals: $6$.
* : Chapter $\text {III}$: Integration: Integration
* : Chapter $10$: Integration: $10.4$. Standard integrals: Other Standard Results: $\text {(xiii)}$
* : Part $\text {III}$: Derivatives and Integrals: Chapter $18$: Integration in Elementary Terms
* : $\S 14$: General Rules of Integration: $14.17$
* : Chapter $25$: Fundamental Integration Formulas: $14$.
* : Chapter $5$. Integrals: Integrals of Special Functions: $9$
* : Appendix: Table $2$: Integrals
* : Appendix: Table $2$: Integrals
* : $\S 16$: Indefinite Integrals: General Rules of Integration: $16.17.$
Category:Primitives involving Secant Function
Category:Tangent Function
Primitive of Square of Secant of a x

== Theorem ==
:$\ds \int \sec^2 a x \rd x = \frac {\tan a x} a + C$
== Proof ==
== Also see ==
* Primitive of $\sin^2 a x$
* Primitive of $\cos^2 a x$
* Primitive of $\tan^2 a x$
* Primitive of $\cot^2 a x$
* Primitive of $\csc^2 a x$
== Sources ==
* : $\S 14$: Integrals involving $\sec a x$: $14.452$
* : Back endpapers: A Brief Table of Integrals: $90$.
* : $\S 17$: Tables of Special Indefinite Integrals: $(22)$ Integrals Involving $\sec a x$: $17.22.2.$
Category:Primitives involving Secant Function
Primitive of Square of Secant of a x over Tangent of a x

== Theorem ==
:$\ds \int \frac {\sec^2 a x \rd x} {\tan a x} = \frac 1 a \ln \size {\tan a x} + C$
== Proof ==
== Also see ==
* Primitive of $\dfrac {\csc^2 a x } {\cot a x}$
== Sources ==
* : $\S 14$: Integrals involving $\tan a x$: $14.433$
Category:Primitives involving Tangent Function
Category:Primitives involving Secant Function
Primitive of Square of Sine Function

== Theorem ==
:$\ds \int \sin^2 x \rd x = \frac x 2 - \frac {\sin 2 x} 4 + C$
where $C$ is an arbitrary constant.
=== Corollary ===
== Proof ==
== Also presented as ==
Some sources present this as:
:$\ds \int \sin^2 x \rd x = \frac 1 2 \paren {x - \frac {\sin 2 x} 2} + C$
== Sources ==
* : Chapter $10$: Integration: $10.4$. Standard integrals: Other Standard Results: $\text {(xxi)}$
* : $\S 14$: General Rules of Integration: $14.21$
* : Appendix: Table $2$: Integrals
* : Appendix: Table $2$: Integrals
* : $\S 16$: Indefinite Integrals: General Rules of Integration: $16.21.$
* : Appendix $7$: Integrals
* : Appendix $8$: Integrals
Category:Primitive of Square of Sine Function
Category:Primitives involving Sine Function
Primitive of Square of Sine Function/Corollary

== Corollary to Primitive of Square of Sine Function ==
:$\ds \int \sin^2 x \rd x = \frac {x - \sin x \cos x} 2 + C$
where $C$ is an arbitrary constant.
== Proof ==
== Sources ==
* : $\S 14$: General Rules of Integration: $14.21$
* : $6$. Integral Calculus: Appendix: Table of Fundamental Standard Integrals
* : $\S 16$: Indefinite Integrals: General Rules of Integration: $16.21.$
Category:Primitive of Square of Sine Function
Primitive of Square of Sine of a x

== Theorem ==
:$\ds \int \sin^2 a x \rd x = \frac x 2 - \frac {\sin 2 a x} {4 a} + C$
== Proof ==
== Also see ==
* Primitive of $\cos^2 a x$
* Primitive of $\tan^2 a x$
* Primitive of $\cot^2 a x$
* Primitive of $\sec^2 a x$
* Primitive of $\csc^2 a x$
== Sources ==
* : $\S 14$: Integrals involving $\sin a x$: $14.347$
* : Front endpapers: A Brief Table of Integrals: $58$.
Category:Primitives involving Sine Function
Primitive of Square of Sine of a x over Cosine of a x

== Theorem ==
:$\ds \int \frac {\sin^2 a x \rd x} {\cos a x} = \frac {-\sin a x} a + \frac 1 a \ln \size {\map \tan {\frac \pi 4 + \frac {a x} 2} } + C$
== Proof ==
== Also see ==
* Primitive of $\dfrac {\cos^2 a x} {\sin a x}$
== Sources ==
* : $\S 14$: Integrals involving $\sin a x$ and $\cos a x$: $14.408$
Category:Primitives involving Sine Function and Cosine Function
Primitive of Square of Tangent Function

== Theorem ==
:$\ds \int \tan^2 x \rd x = \tan x - x + C$
where $C$ is an arbitrary constant.
== Proof 1 ==
== Proof 2 ==
== Sources ==
* : $\S 14$: General Rules of Integration: $14.19$
* : $6$. Integral Calculus: Appendix: Table of Fundamental Standard Integrals
* : $\S 16$: Indefinite Integrals: General Rules of Integration: $16.19.$
Category:Primitive of Square of Tangent Function
Category:Primitives involving Tangent Function
Primitive of Square of Tangent of a x

== Theorem ==
:$\ds \int \tan^2 a x \rd x = \frac {\tan a x} a - x + C$
== Proof ==
== Also see ==
* Primitive of $\sin^2 a x$
* Primitive of $\cos^2 a x$
* Primitive of $\cot^2 a x$
* Primitive of $\sec^2 a x$
* Primitive of $\csc^2 a x$
== Sources ==
* : $\S 14$: Integrals involving $\tan a x$: $14.430$
* : Back endpapers: A Brief Table of Integrals: $84$.
* : $\S 17$: Tables of Special Indefinite Integrals: $(20)$ Integrals Involving $\tan a x$: $17.20.2.$
Category:Primitives involving Tangent Function
Primitive of Tangent Function

== Theorem ==
=== Secant Form ===
=== Cosine Form ===
== Also see ==
* Primitive of Sine Function
* Primitive of Cosine Function
* Primitive of Cotangent Function
* Primitive of Secant Function
* Primitive of Cosecant Function
== Sources ==
* : $\S 14$: General Rules of Integration: $14.13$
* : 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
Category:Primitives of Trigonometric Functions
Category:Primitives involving Tangent Function
Category:Tangent Function