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

== Theorem ==
=== Inverse Hyperbolic Cosine Form ===
=== Logarithm Form ===
== Also see ==
* Primitive of $\dfrac {\sqrt {x^2 + a^2} } {x^2}$
* Primitive of $\dfrac {\sqrt {a^2 - x^2} } {x^2}$
Category:Primitive of Root of x squared minus a squared over x squared
Category:Primitives involving Root of x squared minus a squared
Primitive of Root of x squared minus a squared over x squared/Inverse Hyperbolic Cosine Form

== Theorem ==
:$\ds \int \frac {\sqrt {x^2 - a^2} } {x^2} \rd x = \arcosh \dfrac x a - \frac {\sqrt {x^2 - a^2} } x + C$
for $x^2 \ge a^2$.
== Proof ==
Let:
== Also see ==
* Primitive of $\dfrac {\sqrt {x^2 + a^2} } {x^2}$
* Primitive of $\dfrac {\sqrt {a^2 - x^2} } {x^2}$
== Sources ==
* : Front endpapers: A Brief Table of Integrals: $43$.
Category:Primitive of Root of x squared minus a squared over x squared
Primitive of Root of x squared minus a squared over x squared/Logarithm Form

== Theorem ==
:$\ds \int \frac {\sqrt {x^2 - a^2} } {x^2} \rd x = \frac {-\sqrt {x^2 - a^2} } x + \ln \size {x + \sqrt {x^2 - a^2} } + C$
for $x^2 \ge a^2$.
== Proof ==
Let:
== Also see ==
* Primitive of $\dfrac {\sqrt {x^2 + a^2} } {x^2}$
* Primitive of $\dfrac {\sqrt {a^2 - x^2} } {x^2}$
== Sources ==
* : $\S 14$: Integrals involving $\sqrt {x^2 - a^2}$: $14.221$
Category:Primitive of Root of x squared minus a squared over x squared
Primitive of Root of x squared plus a squared

== Theorem ==
=== Inverse Hyperbolic Sine Form ===
=== Logarithm Form ===
== Also see ==
* Primitive of $\sqrt {x^2 - a^2}$
* Primitive of $\sqrt {a^2 - x^2}$
Category:Primitives involving Root of x squared plus a squared
Category:Primitive of Root of x squared plus a squared
Primitive of Root of x squared plus a squared/Inverse Hyperbolic Sine Form

== Theorem ==
:$\ds \int \sqrt {x^2 + a^2} \rd x = \frac {x \sqrt {x^2 + a^2} } 2 + \frac {a^2} 2 \sinh^{-1} \frac x a + C$
== Proof ==
Let:
Also:
and:
Thus:
== Sources ==
* : Front endpapers: A Brief Table of Integrals: $21$.
* : Appendix: Table $2$: Integrals
* : Appendix: Table $2$: Integrals
Category:Primitive of Root of x squared plus a squared
Primitive of Root of x squared plus a squared/Logarithm Form

== Theorem ==
:$\ds \int \sqrt {x^2 + a^2} \rd x = \frac {x \sqrt {x^2 + a^2} } 2 + \frac {a^2} 2 \map \ln {x + \sqrt {x^2 + a^2} } + C$
== Proof ==
== Sources ==
* : $3$: Elementary Analytic Methods: $3.3$ Rules for Differentiation and Integration: Integrals of Irrational Algebraic Functions: $3.3.41$
* : $\S 14$: Integrals involving $\sqrt {x^2 + a^2}$: $14.189$
* : Chapter $25$: Fundamental Integration Formulas: $26$.
* : $6$. Integral Calculus: Appendix: Table of Fundamental Standard Integrals
* : Appendix $7$: Integrals
* : Appendix $8$: Integrals
Category:Primitive of Root of x squared plus a squared
Primitive of Root of x squared plus a squared cubed

== Theorem ==
:$\ds \int \paren {\sqrt {x^2 + a^2} }^3 \rd x = \frac {x \paren {\sqrt {x^2 + a^2} }^3} 4 + \frac {3 a^2 x \sqrt {x^2 + a^2} } 8 + \frac {3 a^4} 8 \map \ln {x + \sqrt {x^2 + a^2} } + C$
== Proof ==
Let:
== Also see ==
* Primitive of $\paren {\sqrt {x^2 - a^2} }^3$
* Primitive of $\paren {\sqrt {a^2 - z^2} }^3$
== Sources ==
* : $\S 14$: Integrals involving $\sqrt {x^2 + a^2}$: $14.203$
Category:Primitives involving Root of x squared plus a squared
Primitive of Root of x squared plus a squared cubed over x

== Theorem ==
:$\ds \int \frac {\paren {\sqrt {x^2 + a^2} }^3} x \rd x = \frac {\paren {\sqrt {x^2 + a^2} }^3} 3 + a^2 \sqrt {x^2 + a^2} - a^3 \map \ln {\frac {a + \sqrt {x^2 + a^2} } x} + C$
== Proof ==
Let:
== Also see ==
* Primitive of $\dfrac {\paren {\sqrt {x^2 - a^2} }^3} x$
* Primitive of $\dfrac {\paren {\sqrt {a^2 - x^2} }^3} x$
== Sources ==
* : $\S 14$: Integrals involving $\sqrt {x^2 + a^2}$: $14.207$
Category:Primitives involving Root of x squared plus a squared
Primitive of Root of x squared plus a squared cubed over x cubed

== Theorem ==
:$\ds \int \frac {\paren {\sqrt {x^2 + a^2} }^3} {x^3} \rd x = \frac {-\paren {\sqrt {x^2 + a^2} }^3} {2 x^2} + \frac {3 \sqrt {x^2 + a^2} } 2 - \frac {3 a} 2 \map \ln {\frac {a + \sqrt {x^2 + a^2} } x} + C$
== Proof ==
Let:
== Also see ==
* Primitive of $\dfrac {\paren {\sqrt {x^2 - a^2} }^3} {x^3}$
* Primitive of $\dfrac {\paren {\sqrt {a^2 - x^2} }^3} {x^3}$
== Sources ==
* : $\S 14$: Integrals involving $\sqrt {x^2 + a^2}$: $14.209$
* : $\S 17$: Tables of Special Indefinite Integrals: $(9)$ Integrals Involving $\sqrt {x^2 + a^2}$: $17.9.28.$
Category:Primitives involving Root of x squared plus a squared
Primitive of Root of x squared plus a squared cubed over x squared

== Theorem ==
:$\ds \int \frac {\paren {\sqrt {x^2 + a^2} }^3} {x^2} \rd x = \frac {-\paren {\sqrt {x^2 + a^2} }^3} x + \frac {3 x \sqrt {x^2 + a^2} } 2 + \frac {3 a^2} 2 \map \ln {x + \sqrt {x^2 + a^2} } + C$
== Proof ==
Let:
== Also see ==
* Primitive of $\dfrac {\paren {\sqrt {x^2 - a^2} }^3} {x^2}$
* Primitive of $\dfrac {\paren {\sqrt {a^2 - x^2} }^3} {x^2}$
== Sources ==
* : $\S 14$: Integrals involving $\sqrt {x^2 + a^2}$: $14.208$
Category:Primitives involving Root of x squared plus a squared
Primitive of Root of x squared plus a squared over x

== Theorem ==
=== Logarithm Form ===
=== Inverse Hyperbolic Sine Form ===
== Also see ==
* Primitive of $\dfrac {\sqrt {x^2 - a^2} } x$
* Primitive of $\dfrac {\sqrt {a^2 - x^2} } x$
Category:Primitive of Root of x squared plus a squared over x
Category:Primitives involving Root of x squared plus a squared
Primitive of Root of x squared plus a squared over x/Inverse Hyperbolic Sine Form

== Theorem ==
:$\ds \int \frac {\sqrt {x^2 + a^2} } x \rd x = \sqrt {x^2 + a^2} - a \arsinh \size {\dfrac a x} + C$
== Proof ==
== Also see ==
* Primitive of $\dfrac {\sqrt {x^2 - a^2} } x$
* Primitive of $\dfrac {\sqrt {a^2 - x^2} } x$
== Sources ==
* : Front endpapers: A Brief Table of Integrals: $23$.
Category:Primitive of Root of x squared plus a squared over x
Primitive of Root of x squared plus a squared over x/Logarithm Form

== Theorem ==
:$\ds \int \frac {\sqrt {x^2 + a^2} } x \rd x = \sqrt {x^2 + a^2} - a \map \ln {\frac {a + \sqrt {x^2 + a^2} } a} + C$
== Proof ==
Let:
== Also see ==
* Primitive of $\dfrac {\sqrt {x^2 - a^2} } x$
* Primitive of $\dfrac {\sqrt {a^2 - x^2} } x$
== Sources ==
* : $\S 14$: Integrals involving $\sqrt {x^2 + a^2}$: $14.193$
Category:Primitive of Root of x squared plus a squared over x
Primitive of Root of x squared plus a squared over x cubed

== Theorem ==
:$\ds \int \frac {\sqrt {x^2 + a^2} } {x^3} \rd x = \frac {-\sqrt {x^2 + a^2} } {2 x^2} - \frac 1 {2 a} \map \ln {\frac {a + \sqrt {x^2 + a^2} } x} + C$
== Proof ==
Let:
== Also see ==
* Primitive of $\dfrac {\sqrt {x^2 - a^2} } {x^3}$
* Primitive of $\dfrac {\sqrt {a^2 - x^2} } {x^3}$
== Sources ==
* : $\S 14$: Integrals involving $\sqrt {x^2 + a^2}$: $14.195$
Category:Primitives involving Root of x squared plus a squared
Primitive of Root of x squared plus a squared over x squared

== Theorem ==
=== Logarithm Form ===
=== Inverse Hyperbolic Sine Form ===
== Also see ==
* Primitive of $\dfrac {\sqrt {x^2 - a^2} } {x^2}$
* Primitive of $\dfrac {\sqrt {a^2 - x^2} } {x^2}$
Category:Primitive of Root of x squared plus a squared over x squared
Category:Primitives involving Root of x squared plus a squared
Primitive of Root of x squared plus a squared over x squared/Inverse Hyperbolic Sine Form

== Theorem ==
:$\ds \int \frac {\sqrt {x^2 + a^2} } {x^2} \rd x = \arsinh \dfrac x a - \frac {\sqrt {x^2 + a^2} } x + C$
== Proof ==
== Also see ==
* Primitive of $\dfrac {\sqrt {x^2 - a^2} } {x^2}$
* Primitive of $\dfrac {\sqrt {a^2 - x^2} } {x^2}$
== Sources ==
* : Front endpapers: A Brief Table of Integrals: $24$.
Category:Primitive of Root of x squared plus a squared over x squared
Primitive of Root of x squared plus a squared over x squared/Logarithm Form

== Theorem ==
:$\ds \int \frac {\sqrt {x^2 + a^2} } {x^2} \rd x = \frac {-\sqrt {x^2 + a^2} } x + \map \ln {x + \sqrt {x^2 + a^2} } + C$
== Proof ==
Let:
== Also see ==
* Primitive of $\dfrac {\sqrt {x^2 - a^2} } {x^2}$
* Primitive of $\dfrac {\sqrt {a^2 - x^2} } {x^2}$
== Sources ==
* : $\S 14$: Integrals involving $\sqrt {x^2 + a^2}$: $14.194$
Category:Primitive of Root of x squared plus a squared over x squared
Primitive of Secant Function

== Theorem ==
=== Secant plus Tangent Form ===
=== Tangent plus Angle Form ===
== Also presented as ==
== Also see ==
* Primitive of Sine Function
* Primitive of Cosine Function
* Primitive of Tangent Function
* Primitive of Cotangent Function
* Primitive of Cosecant Function
* Primitive of $\dfrac 1 {\cos a x}$
* Primitive of $\dfrac 1 {\sin a x}$
== Sources ==
* : Chapter $\text {III}$: Integration: Integration
* : $\text {II}$. Calculus: Integration
* : Chapter $10$: Integration: $10.4$. Standard integrals: Other Standard Results: $\text {(xx)}$
* : $\S 14$: General Rules of Integration: $14.15$
* : Chapter $5$. Integrals: Integrals of Special Functions: $7$
* : $\S 16$: Indefinite Integrals: General Rules of Integration: $16.15.$
Category:Primitive of Secant Function
Category:Primitives of Trigonometric Functions
Category:Primitives involving Secant Function
Category:Secant Function
Primitive of Secant Function/Secant plus Tangent Form

== Theorem ==
:$\ds \int \sec x \rd x = \ln \size {\sec x + \tan x} + C$
where $\sec x + \tan x \ne 0$.
== Proof 1 ==
== Proof 2 ==
== Also see ==
* Primitive of Sine Function
* Primitive of Cosine Function
* Primitive of Tangent Function
* Primitive of Cotangent Function
* Primitive of Cosecant Function
== Sources ==
* : Chapter $\text {III}$: Integration: Integration
* : $\text {II}$. Calculus: Integration
* : Chapter $10$: Integration: $10.4$. Standard integrals: Other Standard Results: $\text {(xx)}$
* : $\S 14$: General Rules of Integration: $14.15$
* : Chapter $25$: Fundamental Integration Formulas: $12$.
* : Chapter $5$. Integrals: Integrals of Special Functions: $7$
* : Appendix $2$: Table of derivatives and integrals of common functions: Trigonometric functions
* : Appendix: Table $2$: Integrals
* : $\S 5.2$
* : Appendix: Table $2$: Integrals
* : $\S 16$: Indefinite Integrals: General Rules of Integration: $16.15.$
* : Appendix $7$: Integrals
* : Appendix $8$: Integrals
Category:Primitive of Secant Function
Primitive of Secant Function/Secant plus Tangent Form/Mistake

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

== Theorem ==
:$\ds \int \sec x \rd x = \ln \size {\map \tan {\frac x 2 + \frac \pi 4} } + C$
== Proof ==
== Sources ==
* : Chapter $\text {III}$: Integration: Integration
* : $\text {II}$. Calculus: Integration
* : Chapter $10$: Integration: $10.4$. Standard integrals: Other Standard Results: $\text {(xx)}$
* : $\S 14$: General Rules of Integration: $14.15$
* : Chapter $5$. Integrals: Integrals of Special Functions: $7$
* : $6$. Integral Calculus: Appendix: Table of Fundamental Standard Integrals
* : $\S 16$: Indefinite Integrals: General Rules of Integration: $16.15.$
Category:Primitive of Secant Function
Primitive of Secant of a x/Secant plus Tangent Form

== Theorem ==
:$\ds \int \sec a x \rd x = \frac 1 a \ln \size {\sec a x + \tan a x} + C$
where $\sec a x + \tan a x \ne 0$.
== Proof ==
== Also see ==
* Primitive of $\csc a x$: Cosecant minus Cotangent Form
== Sources ==
* : $\S 14$: Integrals involving $\sec a x$: $14.451$
* : Back endpapers: A Brief Table of Integrals: $88$.
* : $\S 17$: Tables of Special Indefinite Integrals: $(22)$ Integrals Involving $\sec a x$: $17.22.1.$
Category:Primitive of Secant of a x
Primitive of Secant of a x/Tangent Form

== Theorem ==
:$\ds \int \sec a x \rd x = \frac 1 a \ln \size {\map \tan {\frac \pi 4 + \frac {a x} 2} } + C$
where $\map \tan {\dfrac \pi 4 + \dfrac {a x} 2} \ne 0$.
== Proof ==
== Also see ==
* Primitive of $\csc a x$: Tangent Form
== Sources ==
* : $\S 14$: Integrals involving $\sec a x$: $14.451$
* : $\S 17$: Tables of Special Indefinite Integrals: $(22)$ Integrals Involving $\sec a x$: $17.22.1.$
Category:Primitive of Secant of a x
Primitive of Secant of a x over x

== Theorem ==
:$\ds \int \frac {\sec a x} x \rd x = \ln \size x + \frac {\paren {a x}^2} 4 + \frac {5 \paren {a x}^4} {96} + \frac {61 \paren {a x}^6} {4320} + \cdots + \frac {\paren {-1}^n E_n \paren {a x}^{2 n} } {\paren {2 n} \paren {2 n}!} + \cdots + C$
where $E_n$ is the $n$th Euler number.
== Proof ==
== Also see ==
* Primitive of $\dfrac {\sin a x} x$
* Primitive of $\dfrac {\cos a x} x$
* Primitive of $\dfrac {\tan a x} x$
* Primitive of $\dfrac {\cot a x} x$
* Primitive of $\dfrac {\csc a x} x$
== Sources ==
* : $\S 14$: Integrals involving $\sec a x$: $14.457$
Category:Primitives involving Secant Function
Primitive of Sine Function

== Theorem ==
:$\ds \int \sin x \rd x = -\cos x + C$
where $C$ is an arbitrary constant.
=== Corollary ===
== Proof ==
== Also see ==
* Primitive of Cosine Function
* Primitive of Tangent Function
* Primitive of Cotangent Function
* Primitive of Secant Function
* Primitive of Cosecant Function
== Sources ==
* : Chapter $\text {III}$: Integration: Integration
* : $\text {II}$. Calculus: Integration
* : Chapter $10$: Integration: $10.4$. Standard integrals: Standard Forms: $\text {(iv)}$
* : $\S 14$: General Rules of Integration: $14.11$
* : Front endpapers: A Brief Table of Integrals: $4$.
* : $6$. 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: $16.11.$
* : Appendix $7$: Integrals
* : Appendix $8$: Integrals
Category:Primitive of Sine Function
Category:Primitives of Trigonometric Functions
Category:Primitives involving Sine Function
Category:Sine Function
Primitive of Sine Function/Corollary

== Corollary to Primitive of Sine Function ==
:$\ds \int \sin a x \rd x = - \frac {\cos a x} a + C$
where $a$ is a non-zero constant.
== Proof ==
== Also see ==
* Primitive of $\cos a x$
* Primitive of $\tan a x$
* Primitive of $\cot a x$
* Primitive of $\sec a x$
* Primitive of $\csc a x$
== Sources ==
* : $\S 14$: Integrals involving $\sin a x$: $14.339$
* : Front endpapers: A Brief Table of Integrals: $56$.
* : $9$ Sums and Integrals: $9.8$ Standard Integrals
* : $\S 17$: Tables of Special Indefinite Integrals: $(17)$ Integrals Involving $\sin a x$: $17.17.1.$
Category:Primitive of Sine Function
Primitive of Sine Function/Proof

== Theorem ==
== Proof ==
From Derivative of Cosine Function:
:$\map {\dfrac \d {\d x} } {-\cos x} = \sin x$
The result follows from the definition of primitive.
== Sources ==
* : Chapter $\text {II}$: Integration of Elementary Functions: $\S 7$. Standard Integrals: $5$.
* : $\text {II}$. Calculus: Integration
* : Part $\text {III}$: Derivatives and Integrals: Chapter $18$: Integration in Elementary Terms
* : Chapter $25$: Fundamental Integration Formulas: $8$.
* : Chapter $5$. Integrals: Integrals of Special Functions: $3$
Category:Primitive of Sine Function
Primitive of Sine Integral Function

== Theorem ==
:$\ds \int \map \Si x \rd x = x \map \Si x + \cos x + C$
where $\Si$ denotes the sine integral function.
== Proof ==
By Derivative of Sine Integral Function, we have: 
:$\map {\dfrac \d {\d x} } {\map \Si x} = \dfrac {\sin x} x$
So:
Category:Primitives
Category:Sine Integral Function
Primitive of Sine of a x + b

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

== Examples of Use of Primitive of $\map \sin {a x + b}$ ==
:$\ds \int \map \sin {3 - x} \rd x = \map \cos {3 - x} + C$
== Proof ==
From Primitive of $\map \sin {a x + b}$:
== Sources ==
* : $\text {II}$. Calculus: Exercises $\text {XIV}$: $1$.
Category:Primitive of Sine of a x + b
Primitive of Sine of a x + b/Examples/3x + 4

== Examples of Use of Primitive of $\map \sin {a x + b}$ ==
:$\ds \int \map \sin {3 x + 4} \rd x = -\dfrac {\map \cos {3 x + 4} } 3 + C$
== Proof ==
An instance of Primitive of $\map \sin {a x + b}$, substituting $3$ for $a$ and $4$ for $b$.
== Sources ==
* : $\text {II}$. Calculus: Integration: Example
Category:Primitive of Sine of a x + b
Primitive of Sine of a x + b/Proof 2

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

== Theorem ==
:$\ds \int \sin a x \cos a x \rd x = \frac {\sin^2 a x} {2 a} + C$
== Proof ==
== Sources ==
* : $\S 14$: Integrals involving $\sin a x$ and $\cos a x$: $14.399$
* : Front endpapers: A Brief Table of Integrals: $63$.
* : $\S 17$: Tables of Special Indefinite Integrals: $(19)$ Integrals Involving $\sin a x$ and $\cos a x$: $17.19.1.$
Category:Primitives involving Sine Function and Cosine Function
Primitive of Sine of a x by Cosine of b x

== Theorem ==
:$\ds \int \sin a x \cos b x \rd x = \frac {-\map \cos {a - b} x} {2 \paren {a - b} } - \frac {\map \cos {a + b} x} {2 \paren {a + b} } + C$
for $a, b \in \R: a \ne b$
== Proof ==
== Also see ==
* Primitive of $\sin a x \cos a x$ for $a = b$
* Primitive of $\sin a x \sin b x$
* Primitive of $\cos a x \cos b x$
== Sources ==
* : $\S 14$: Integrals involving $\sin a x$ and $\cos a x$: $14.400$
* : Front endpapers: A Brief Table of Integrals: $62. \ \text {(a)}$
* : $\S 17$: Tables of Special Indefinite Integrals: $(19)$ Integrals Involving $\sin a x$ and $\cos a x$: $17.19.2.$
Category:Primitives involving Sine Function and Cosine Function
Primitive of Sine of a x by Sine of b x

== Theorem ==
For $p \ne q$:
:$\ds \int \sin a x \sin b x \rd x = \frac {\map \sin {a - b} x} {2 \paren {a - b} } - \frac {\map \sin {a + b} x} {2 \paren {a + b} } + C$
== Proof ==
== Also see ==
* Primitive of $\cos a x \cos p x$
* Primitive of $\sin a x \cos b x$
* For $a = b$, see Primitive of $\sin^2 a x$
== Sources ==
* : $\S 14$: Integrals involving $\sin a x$: $14.353$
* : Front endpapers: A Brief Table of Integrals: $62. \ \text {(b)}$
Category:Primitives involving Sine Function
Primitive of Sine of a x over Cosine of a x

== Theorem ==
:$\ds \int \dfrac {\sin a x} {\cos a x} \rd x = -\dfrac 1 a \ln \size {\cos a x} + C$
== Proof ==
== Also see ==
* Primitive of $\dfrac {\cos a x} {\sin a x}$
== Sources ==
* : Front endpapers: A Brief Table of Integrals: $67$.
Category:Primitives involving Sine Function and Cosine Function
Primitive of Sine of a x over Power of p plus q of Cosine of a x

== Theorem ==
:$\ds \int \frac {\sin a x \rd x} {\paren {p + q \cos a x}^n} = \frac 1 {a q \paren {n - 1} \paren {p + q \cos a x}^{n - 1} } + C$
== Proof ==
== Also see ==
* Primitive of $\dfrac {\cos a x} {\paren {p + q \sin a x}^n}$
== Sources ==
* : $\S 14$: Integrals involving $\sin a x$ and $\cos a x$: $14.417$
Category:Primitives involving Sine Function and Cosine Function
Primitive of Sine of a x over Power of x

== Theorem ==
:$\ds \int \frac {\sin a x} {x^n} \rd x = \frac {-\sin a x} {\paren {n - 1} x^{n - 1} } + \frac a {n - 1} \int \frac {\cos a x} {x^{n - 1} } \rd x$
== Proof ==
With a view to expressing the primitive in the form:
:$\ds \int u \frac {\rd v} {\rd x} \rd x = u v - \int v \frac {\rd u} {\rd x} \rd x$
let:
and let:
Then:
== Also see ==
* Primitive of $\dfrac {\cos a x} {x^n}$
== Sources ==
* : $\S 14$: Integrals involving $\sin a x$: $14.365$
Category:Primitives involving Sine Function
Primitive of Sine of a x over Sine of a x minus Cosine of a x

== Theorem ==
:$\ds \int \frac {\sin a x \rd x} {\sin a x - \cos a x} = \frac x 2 + \frac 1 {2 a} \ln \size {\sin a x - \cos a x} + C$
== Proof ==
First note that:
Then:
== Also see ==
* Primitive of $\dfrac {\sin a x} {\sin a x + \cos a x}$
* Primitive of $\dfrac {\cos a x} {\sin a x + \cos a x}$
* Primitive of $\dfrac {\cos a x} {\sin a x - \cos a x}$
== Sources ==
* : $\S 14$: Integrals involving $\sin a x$ and $\cos a x$: $14.413$
Category:Primitives involving Sine Function and Cosine Function
Primitive of Sine of a x over Sine of a x plus Cosine of a x

== Theorem ==
:$\ds \int \frac {\sin a x \rd x} {\sin a x + \cos a x} = \frac x 2 - \frac 1 {2 a} \ln \size {\sin a x + \cos a x} + C$
== Proof ==
First note that:
Then:
== Also see ==
* Primitive of $\dfrac {\sin a x} {\sin a x - \cos a x}$
* Primitive of $\dfrac {\cos a x} {\sin a x + \cos a x}$
* Primitive of $\dfrac {\cos a x} {\sin a x - \cos a x}$
== Sources ==
* : $\S 14$: Integrals involving $\sin a x$ and $\cos a x$: $14.413$
Category:Primitives involving Sine Function and Cosine Function
Primitive of Sine of a x over Sine of a x plus phi

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

== Theorem ==
:$\ds \int \frac {\sin a x \rd x} {p + q \cos a x} = \frac {-1} {a q} \ln \size {p + q \cos a x} + C$
== Proof ==
== Also see ==
* Primitive of $\dfrac {\cos a x} {p + q \sin a x}$
== Sources ==
* : $\S 14$: Integrals involving $\sin a x$ and $\cos a x$: $14.415$
Category:Primitives involving Sine Function and Cosine Function
Primitive of Sine of a x over x

== Theorem ==
== Proof ==
The validity of $(1)$ follows from Sine Function is Absolutely Convergent.
== Also see ==
* Primitive of $\dfrac {\cos a x} x$
* Primitive of $\dfrac {\tan a x} x$
* Primitive of $\dfrac {\cot a x} x$
* Primitive of $\dfrac {\sec a x} x$
* Primitive of $\dfrac {\csc a x} x$
* Primitive of $\dfrac {\sin x} x$ has no Solution in Elementary Functions
== Sources ==
* : $\S 14$: Integrals involving $\sin a x$: $14.343$
Category:Primitives involving Sine Function
Primitive of Sine of a x over x squared

== Theorem ==
:$\ds \int \frac {\sin a x \rd x} {x^2} = -\frac {\sin a x} x + a \int \frac {\cos a x \rd x} x + C$
== 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 $\dfrac {\cos a x} x$
* Primitive of $\dfrac {\cos a x} {x^2}$
== Sources ==
* : $\S 14$: Integrals involving $\sin a x$: $14.344$
Category:Primitives involving Sine Function
Primitive of Sine of a x squared by Cosine of a x squared

== Theorem ==
:$\ds \int \sin^2 a x \cos^2 a x \rd x = \frac x 8 - \frac {\sin 4 a x} {32 a} + C$
== Proof 1 ==
== Proof 2 ==
== Sources ==
* : $\S 14$: Integrals involving $\sin a x$ and $\cos a x$: $14.403$
Category:Primitives involving Sine Function and Cosine Function
Category:Primitive of Sine of a x squared by Cosine of a x squared