nyuuzyou/wikis huggingface dataset:
Primitive of Reciprocal of Root of x squared minus a squared/Inverse Hyperbolic Cosine Form

== Theorem ==
:$\ds \int \frac {\d x} {\sqrt {x^2 - a^2} } = \dfrac {\size x} x \arcosh {\size {\frac x a} } + C$
for $x^2 > a^2$.
== Proof ==
When $x = a$ we have that $\sqrt {x^2 - a^2} = 0$ and then $\dfrac 1 {\sqrt {x^2 - a^2} }$ is not defined.
When $\size x < a$ we have that $x^2 - a^2 < 0$ and then $\sqrt {x^2 - a^2}$ is not defined.
Hence the domain needs to be restricted to $\size x > a$, or that is: $\size {\dfrac x a} > 1$.
Let $x > a$.
Let:
Let $x < -a$.
Let $z = -x$.
Hence:
:$\dfrac {\d z} {\d x} = -1$
Then:
== Also see ==
* Primitive of $\dfrac 1 {\sqrt {x^2 + a^2} }$: Inverse Hyperbolic Sine Form
* Primitive of $\dfrac 1 {\sqrt {a^2 - x^2} }$
== Sources ==
* : Chapter $\text {II}$: Integration of Elementary Functions: $\S 7$. Standard Integrals: $15$.
:::who glosses over the details of what happens for negative $x$.
* : Chapter $10$: Integration: $10.4$. Standard integrals: Standard Forms: $\text {(xi)}$
:::who covers positive $x$ only.
* : Front endpapers: A Brief Table of Integrals: $36$.
:::who neglects the case of negative $x$.
* : $6$. Integral Calculus: Appendix: Table of Fundamental Standard Integrals
:::who neglects the case of negative $x$.
* : Appendix: Table $2$: Integrals
:::who neglects the case of negative $x$.
* : Appendix: Table $2$: Integrals
:::who neglects the case of negative $x$.
* : Appendix $7$: Integrals
:::who covers positive $x$ only
* : Appendix $8$: Integrals
:::who covers positive $x$ only
Category:Expressions whose Primitives are Inverse Hyperbolic Functions
Category:Primitives involving Inverse Hyperbolic Cosine Function‎
Category:Primitive of Reciprocal of Root of x squared minus a squared
Primitive of Reciprocal of Root of x squared minus a squared/Inverse Hyperbolic Cosine Form/Mistake

== Source Work ==
:Front endpapers: A Brief Table of Integrals: $36$.
:$6$. Integral Calculus: Appendix: Table of Fundamental Standard Integrals
:Table 2 Integrals
== Mistake ==
:$\ds \int \frac {\d x} {\sqrt {x^2 - a^2} } = \cosh^{-1} \frac a x + C$
== Correction ==
Should be:
:$\ds \int \frac {\d x} {\sqrt {x^2 - a^2} } = \dfrac {\size x} x \arcosh {\size {\frac x a} } + C$
or an equivalent form.
== Sources ==
* : Front endpapers: A Brief Table of Integrals: $36$.
* : $6$. Integral Calculus: Appendix: Table of Fundamental Standard Integrals
* : Appendix: Table $2$: Integrals
* : Appendix: Table $2$: Integrals
Category:Mistakes/Calculus and Analytic Geometry
Category:Mistakes/Mathematics for Engineers and Scientists
Category:Mistakes/The Penguin Dictionary of Mathematics
Primitive of Reciprocal of Root of x squared minus a squared/Logarithm Form

== Theorem ==
:$\ds \int \frac {\d x} {\sqrt {x^2 - a^2} } = \ln \size {x + \sqrt {x^2 - a^2} } + C$
for $0 < a < \size x$.
=== Corollary ===
== Proof ==
We have that $\sqrt {x^2 - a^2}$ is defined only when $x^2 > a^2$, that is, either:
:$x > a$
or:
:$x < -a$
where it is assumed that $a > 0$.
Consider the arcsecant substitution:
:$u = \arcsec {\dfrac x a}$
which is defined for all $x$ such that $\size {\dfrac x a} \ge 1$.
That is:
:$\size x \ge a$
Hence from Shape of Secant Function, this substitution is valid for all for all $x$ such that $\size {\dfrac x a} > 1$.
Let $x > a$.
Then:
Now suppose $x < -a$.
Let $z = -x$.
Then:
:$\d x = -\d z$
and we then have:
The result follows.
== Also presented as ==
== Also see ==
* Primitive of $\dfrac 1 {\sqrt {x^2 - a^2} }$: Logarithm Form
* Primitive of $\dfrac 1 {\sqrt {a^2 - x^2} }$
== Sources ==
* : Chapter $\text {II}$: Integration of Elementary Functions: $\S 7$. Standard Integrals: $15$.
* : $3$: Elementary Analytic Methods: $3.3$ Rules for Differentiation and Integration: Integrals of Irrational Algebraic Functions: $3.3.40$
* : $\S 14$: General Rules of Integration: $14.44$
* : $\S 14$: Integrals involving $\sqrt {x^2 - a^2}$: $14.210$
* : Front endpapers: A Brief Table of Integrals: $36$.
* : Chapter $25$: Fundamental Integration Formulas: $24$.
* : Chapter $5$. Integrals: Integrals of Special Functions: $26$
* : $\S 16$: Indefinite Integrals: General Rules of Integration: $16.44.$
* : $\S 17$: Tables of Special Indefinite Integrals: $(10)$ Integrals Involving $\sqrt {x^2 - a^2}$: $17.10.1.$
* : Appendix $7$: Integrals
* : Appendix $8$: Integrals
Category:Primitive of Reciprocal of Root of x squared minus a squared
Primitive of Reciprocal of Root of x squared minus a squared/Logarithm Form/Also presented as

== Primitive of $\frac 1 {\sqrt {x^2 - a^2} }$: Logarithm Form: Also presented as ==
The standard presentation of this result is:
Some sources present this in the form:
:$\ds \int \frac {\d x} {\sqrt {x^2 - a^2} } = \ln \size {\dfrac {x + \sqrt {x^2 - a^2} } a} + C$
which is the same as above, except that the constant $a$ has not been subsumed into the arbitrary constant $C$.
== Sources ==
* : Chapter $\text {III}$: Integration: Integration
* : Chapter $10$: Integration: $10.4$. Standard integrals: Standard Forms: $\text {(xi)}$
* : $6$. Integral Calculus: Appendix: Table of Fundamental Standard Integrals
Category:Primitive of Reciprocal of Root of x squared minus a squared
Primitive of Reciprocal of Root of x squared minus a squared/Logarithm Form/Corollary

== Theorem ==
:$\ds \int \frac {\d x} {-\sqrt {x^2 - a^2} } = \ln \size {x - \sqrt {x^2 - a^2} } + C$
for $\size x > a$.
== Proof ==
== Also see ==
* Primitive of $\dfrac 1 {\sqrt {x^2 + a^2} }$: Logarithm Form
* Primitive of $\dfrac 1 {\sqrt {a^2 - x^2} }$
== Sources ==
* : Chapter $\text {II}$: Integration of Elementary Functions: $\S 7$. Standard Integrals: $15$.
Category:Primitive of Reciprocal of Root of x squared minus a squared
Primitive of Reciprocal of Root of x squared minus a squared cubed

== Theorem ==
:$\ds \int \frac {\d x} {\paren {\sqrt {x^2 - a^2} }^3} = \frac {-x} {a^2 \sqrt {x^2 - a^2} } + C$
for $\size x > a$.
== Proof ==
Let:
== Also see ==
* Primitive of $\dfrac 1 {\paren {\sqrt {x^2 + a^2} }^3}$
* Primitive of $\dfrac 1 {\paren {\sqrt {a^2 - x^2} }^3}$
== Sources ==
* : $\S 14$: Integrals involving $\sqrt {x^2 - a^2}$: $14.223$
Category:Primitives involving Root of x squared minus a squared
Primitive of Reciprocal of Root of x squared plus Constant

== Theorem ==
:$\ds \int \frac {\d x} {\sqrt {x^2 + k} } = \ln \size {x + \sqrt {x^2 + k} } + C$
== Proof ==
=== Positive Constant ===
Let $k > 0$.
Then $k = a^2$ for some $a \in \R$.
Hence from Primitive of $\dfrac 1 {\sqrt {x^2 + a^2} }$: Logarithm Form:
from which the result follows.
=== Negative Constant ===
Let $k < 0$.
Then $k = -a^2$ for some $a \in \R$.
Hence from Primitive of $\dfrac 1 {\sqrt {x^2 - a^2} }$: Logarithm Form:
from which the result follows.
=== Zero Constant ===
Let $k = 0$.
Then we have:
The result follows.
== Sources ==
* : Chapter $\text {III}$: Integration: Integration
Category:Primitive of Reciprocal of Root of x squared minus a squared
Primitive of Reciprocal of Root of x squared plus a squared

== Theorem ==
=== Inverse Hyperbolic Sine Form ===
=== Logarithm Form ===
== Examples ==
== Also see ==
* Primitive of $\dfrac 1 {\sqrt {x^2 - a^2} }$
* Primitive of $\dfrac 1 {\sqrt {a^2 - x^2} }$
Category:Primitive of Reciprocal of Root of x squared plus a squared
Category:Primitives involving Root of x squared plus a squared
Category:Primitives involving Reciprocals
Primitive of Reciprocal of Root of x squared plus a squared/Examples

== Examples of Use of Primitive of $\dfrac 1 {\sqrt {x^2 + a^2} }$ ==
=== Primitive of $\dfrac 1 {\sqrt {16 + 9 x^2} }$ ===
Category:Examples of Use of Primitive of Reciprocal of Root of x squared plus a squared
Primitive of Reciprocal of Root of x squared plus a squared/Examples/16 + 9 x^2

== Example of Use of Primitive of $\dfrac 1 {\sqrt {x^2 + a^2} }$ ==
:$\ds \int \dfrac 1 {\sqrt {16 + 9 x^2} } = \dfrac 1 3 \map \ln {3 x + \sqrt {9 x^2 + 16} }$
== Proof ==
We have from Primitive of $\dfrac 1 {\sqrt {x^2 + a^2} }$:
Let $y = 3 x$.
Then:
== Sources ==
* : $\text {II}$. Calculus: Exercises $\text {XIV}$: $14$.
Category:Examples of Use of Primitive of Reciprocal of Root of x squared plus a squared
Primitive of Reciprocal of Root of x squared plus a squared/Inverse Hyperbolic Sine Form

== Theorem ==
:$\ds \int \frac {\d x} {\sqrt {x^2 + a^2} } = \arsinh {\frac x a} + C$
== Proof ==
Let:
== Also see ==
* Primitive of $\dfrac 1 {\sqrt {x^2 - a^2} }$: Inverse Hyperbolic Cosine Form
* Primitive of $\dfrac 1 {\sqrt {a^2 - x^2} }$
== Sources ==
* : Chapter $\text {II}$: Integration of Elementary Functions: $\S 7$. Standard Integrals: $14$.
* : Chapter $\text {III}$: Integration: Integration
* : Chapter $10$: Integration: $10.4$. Standard integrals: Standard Forms: $\text {(x)}$
* : $\S 14$: General Rules of Integration: $14.43$
* : $\S 14$: Integrals involving $\sqrt {x^2 + a^2}$: $14.182$
* : Front endpapers: A Brief Table of Integrals: $20$.
* : $6$. Integral Calculus: Appendix: Table of Fundamental Standard Integrals
* : Appendix: Table $2$: Integrals
* : Appendix: Table $2$: Integrals
* : $\S 16$: Indefinite Integrals: General Rules of Integration: $16.43.$
* : $\S 17$: Tables of Special Indefinite Integrals: $(9)$ Integrals Involving $\sqrt {x^2 + a^2}$: $17.9.1.$
* : Appendix $7$: Integrals
* : Appendix $8$: Integrals
Category:Primitive of Reciprocal of Root of x squared plus a squared
Primitive of Reciprocal of Root of x squared plus a squared/Logarithm Form

== Theorem ==
:$\ds \int \frac {\d x} {\sqrt {x^2 + a^2} } = \map \ln {x + \sqrt {x^2 + a^2} } + C$
=== Corollary ===
== Proof 1 ==
== Proof 2 ==
== Also presented as ==
== Also see ==
* Primitive of $\dfrac 1 {\sqrt {x^2 - a^2} }$: Logarithm Form
* Primitive of $\dfrac 1 {\sqrt {a^2 - x^2} }$
== Sources ==
* : Chapter $\text {II}$: Integration of Elementary Functions: $\S 7$. Standard Integrals: $14$.
* : $\text {II}$. Calculus: Integration
* : $3$: Elementary Analytic Methods: $3.3$ Rules for Differentiation and Integration: Integrals of Irrational Algebraic Functions: $3.3.40$
* : $\S 14$: General Rules of Integration: $14.43$
* : $\S 14$: Integrals involving $\sqrt {x^2 + a^2}$: $14.182$
* : Chapter $25$: Fundamental Integration Formulas: $23$.
* : Chapter $5$. Integrals: Integrals of Special Functions: $26$
* : $\S 16$: Indefinite Integrals: General Rules of Integration: $16.43.$
* : $\S 17$: Tables of Special Indefinite Integrals: $(9)$ Integrals Involving $\sqrt {x^2 + a^2}$: $17.9.1.$
* : Appendix $7$: Integrals
* : Appendix $8$: Integrals
Category:Primitive of Reciprocal of Root of x squared plus a squared
Primitive of Reciprocal of Root of x squared plus a squared/Logarithm Form/Also presented as

== Primitive of $\frac 1 {\sqrt {x^2 + a^2} }$: Logarithm Form: Also presented as ==
The standard presentation of this result is:
Some sources present this in the form:
:$\ds \int \frac {\d x} {\sqrt {x^2 + a^2} } = \map \ln {\dfrac {x + \sqrt {x^2 + a^2} } a} + C$
which is the same as above, except that the constant $a$ has not been subsumed into the arbitrary constant $C$.
Some sources present it as:
:$\ds \int \frac {\d x} {\sqrt {x^2 + a^2} } = \ln \size {x + \sqrt {x^2 + a^2} } + C$
== Sources ==
* : Chapter $\text {III}$: Integration: Integration
* : Chapter $10$: Integration: $10.4$. Standard integrals: Standard Forms: $\text {(x)}$
* : Front endpapers: A Brief Table of Integrals: $20$.
Category:Primitive of Reciprocal of Root of x squared plus a squared
Primitive of Reciprocal of Root of x squared plus a squared/Logarithm Form/Corollary

== Theorem ==
:$\ds \int \frac {\d x} {-\sqrt {x^2 + a^2} } = \ln \size {x - \sqrt {x^2 + a^2} } + C$
== Proof ==
== Also see ==
* Primitive of $\dfrac 1 {\sqrt {x^2 - a^2} }$: Logarithm Form
* Primitive of $\dfrac 1 {\sqrt {a^2 - x^2} }$
== Sources ==
* : Chapter $\text {II}$: Integration of Elementary Functions: $\S 7$. Standard Integrals: $14$.
Category:Primitive of Reciprocal of Root of x squared plus a squared
Primitive of Reciprocal of Root of x squared plus a squared/Logarithm Form/Proof 2

== Theorem ==
== Proof ==
Let $y^2 = a^2 + x^2$.
Then:
== Sources ==
* : $\text {II}$. Calculus: Integration: Algebraic Integration: $\text {IV}$.
Category:Primitive of Reciprocal of Root of x squared plus a squared
Primitive of Reciprocal of Root of x squared plus a squared/Mistake

== Source Work ==
:
:Appendix $7$: Integrals
== Mistake ==
:$\ds \int \frac {\d x} {\sqrt {a^2 + x^2} } = \dfrac 1 a \tan^{-1} {\frac x a}$
== Correction ==
This should read:
:$\ds \int \frac {\d x} {a^2 + x^2} = \dfrac 1 a \tan^{-1} {\frac x a}$
This has been corrected in the $6$th edition.
== Sources ==
* : Appendix $7$: Integrals
Category:Mistakes/The Concise Oxford Dictionary of Mathematics
Primitive of Reciprocal of Root of x squared plus a squared cubed

== Theorem ==
:$\ds \int \frac {\d x} {\paren {\sqrt {x^2 + a^2} }^3} = \frac x {a^2 \sqrt {x^2 + a^2} } + C$
== Proof 1 ==
== Proof 2 ==
== Also see ==
* Primitive of $\dfrac 1 {\paren {\sqrt {x^2 - a^2} }^3}$
* Primitive of $\dfrac 1 {\paren {\sqrt {a^2 - x^2} }^3}$
== Sources ==
* : $\S 14$: Integrals involving $\sqrt {x^2 + a^2}$: $14.196$
Category:Primitives involving Root of x squared plus a squared
Category:Primitive of Reciprocal of Root of x squared plus a squared cubed
Primitive of Reciprocal of Root of x squared plus k

== Theorem ==
Let $k \in \R$.
Then:
:$\ds \int \frac {\d x} {\sqrt {\size {x^2 + k} } } = \map \ln {x + \sqrt {\size {x^2 + k} } } + C$
== Proof ==
There are three cases:
:$0 \le k$
:$-x^2 < k < 0$
:$k < -x^2$
=== $(1): \quad 0 \le k$ ===
If $0 \le k$ then $k = a^2$ for some $a \in \R$.
Then Primitive of $\dfrac 1 {\sqrt {x^2 + a^2} }$ applies:
:$\ds \int \frac {\d x} {\sqrt {x^2 + a^2} } = \map \ln {x + \sqrt {x^2 + a^2} } + C$
=== $(2): \quad -x^2 < k < 0$ ===
If $-x^2 < k < 0$ then:
:$k = -a^2$ for some $a \in \R$
and:
:$x^2 - a^2 > 0$
Then Primitive of $\dfrac 1 {\sqrt {x^2 - a^2} }$ applies:
:$\ds \int \frac {\d x} {\sqrt {x^2 - a^2} } = \map \ln {x + \sqrt {x^2 - a^2} } + C$
=== $(3): \quad k < -x^2$ ===
If $k < -x^2$ then:
:$k = -a^2$ for some $a \in \R$
and:
:$a^2 - x^2 > 0$
Then Primitive of $\dfrac 1 {\sqrt {a^2 - x^2} }$ applies:
:$\ds \int \frac {\d x} {\sqrt {a^2 - x^2} } = \map \ln {x + \sqrt {a^2 - x^2} } + C$
The result holds for all three cases.
Category:Primitives involving Root of x squared plus a squared
Category:Primitives involving Root of x squared minus a squared
Category:Primitives involving Root of a squared minus x squared
Category:Primitives involving Reciprocals
Primitive of Reciprocal of Secant of a x

== Theorem ==
:$\ds \int \frac {\d x} {\sec a x} = \frac {\sin a x} a + C$
== Proof ==
== Also see ==
* Primitive of $\dfrac 1 {\sin a x}$
* Primitive of $\dfrac 1 {\cos a x}$
* Primitive of $\dfrac 1 {\tan a x}$
* Primitive of $\dfrac 1 {\cot a x}$
* Primitive of $\dfrac 1 {\csc a x}$
== Sources ==
* : $\S 14$: Integrals involving $\sec a x$: $14.455$
Category:Primitives involving Secant Function
Primitive of Reciprocal of Sine of a x

== Theorem ==
=== Logarithm of Cosecant minus Cotangent Form ===
=== Logarithm of Tangent Form ===
== Also see ==
* Primitive of $\dfrac 1 {\cos a x}$
* Primitive of $\dfrac 1 {\tan a x}$
* Primitive of $\dfrac 1 {\cot a x}$
* Primitive of $\dfrac 1 {\sec a x}$
* Primitive of $\dfrac 1 {\csc a x}$
Category:Primitives involving Sine Function
Category:Primitive of Reciprocal of Sine of a x
Primitive of Reciprocal of Sine of a x/Logarithm of Cosecant minus Cotangent Form

== Theorem ==
:$\ds \int \frac {\d x} {\sin a x} = \frac 1 a \ln \size {\csc a x - \cot a x} + C$
== Proof ==
== Also see ==
* Primitive of $\dfrac 1 {\cos a x}$: $\sec + \tan$ form
== Sources ==
* : $\S 14$: Integrals involving $\sin a x$: $14.345$
Category:Primitive of Reciprocal of Sine of a x
Primitive of Reciprocal of Sine of a x/Logarithm of Tangent Form

== Theorem ==
:$\ds \int \frac {\d x} {\sin a x} = \frac 1 a \ln \size {\tan \frac {a x} 2} + C$
== Proof ==
== Also see ==
* Primitive of $\dfrac 1 {\cos a x}$: $\tan$ form
== Sources ==
* : $\S 14$: Integrals involving $\sin a x$: $14.345$
Category:Primitive of Reciprocal of Sine of a x
Primitive of Reciprocal of Sine of a x by 1 minus Cosine of a x

== Theorem ==
:$\ds \int \frac {\d x} {\sin a x \paren {1 - \cos a x} } = \frac {-1} {2 a \paren {1 - \cos a x} } + \frac 1 {2 a} \ln \size {\tan \frac {a x} 2} + C$
== Proof ==
Let:
Then:
== Also see ==
* Primitive of $\dfrac 1 {\sin a x \paren {1 + \cos a x} }$
* Primitive of $\dfrac 1 {\cos a x \paren {1 + \sin a x} }$
* Primitive of $\dfrac 1 {\cos a x \paren {1 - \sin a x} }$
== Sources ==
* : $\S 14$: Integrals involving $\sin a x$ and $\cos a x$: $14.411$
Category:Primitives involving Sine Function and Cosine Function
Primitive of Reciprocal of Sine of a x by 1 plus Cosine of a x

== Theorem ==
:$\ds \int \frac {\d x} {\sin a x \paren {1 + \cos a x} } = \frac 1 {2 a \paren {1 + \cos a x} } + \frac 1 {2 a} \ln \size {\tan \frac {a x} 2} + C$
== Proof ==
Let:
Then:
== Also see ==
* Primitive of $\dfrac 1 {\sin a x \paren {1 - \cos a x} }$
* Primitive of $\dfrac 1 {\cos a x \paren {1 + \sin a x} }$
* Primitive of $\dfrac 1 {\cos a x \paren {1 - \sin a x} }$
== Sources ==
* : $\S 14$: Integrals involving $\sin a x$ and $\cos a x$: $14.411$
Category:Primitives involving Sine Function and Cosine Function
Primitive of Reciprocal of Sine of a x by Square of Cosine of a x

== Theorem ==
:$\ds \int \frac {\d x} {\sin a x \cos^2 a x} = \frac 1 a \ln \size {\tan \frac {a x} 2} + \frac 1 {a \cos a x} + C$
== Proof ==
== Also see ==
* Primitive of $\dfrac 1 {\sin^2 a x \cos a x}$
== Sources ==
* : $\S 14$: Integrals involving $\sin a x$ and $\cos a x$: $14.406$
Category:Primitives involving Sine Function and Cosine Function
Primitive of Reciprocal of Sine of a x minus Cosine of a x

== Theorem ==
:$\ds \int \frac {\d x} {\sin a x - \cos a x} = \frac 1 {a \sqrt 2} \ln \size {\map \tan {\frac {a x} 2 - \frac \pi 8} } + C$
== Proof ==
Let:
== Also see ==
* Primitive of $\dfrac 1 {\sin a x + \cos a x}$
== Sources ==
* : $\S 14$: Integrals involving $\sin a x$ and $\cos a x$: $14.412$
Category:Primitives involving Sine Function and Cosine Function
Primitive of Reciprocal of Sine of a x plus Cosine of a x

== Theorem ==
:$\ds \int \frac {\d x} {\sin a x + \cos a x} = \frac 1 {a \sqrt 2} \ln \size {\map \tan {\frac {a x} 2 + \frac \pi 8} } + C$
== Proof 1 ==
== Proof 2 ==
== Examples ==
== Also see ==
* Primitive of $\dfrac 1 {\sin a x - \cos a x}$
== Sources ==
* : $\S 14$: Integrals involving $\sin a x$ and $\cos a x$: $14.412$
Category:Primitive of Reciprocal of Sine of a x plus Cosine of a x
Category:Primitives involving Sine Function and Cosine Function
Primitive of Reciprocal of Sine of a x plus Cosine of a x/Examples

== Examples of Use of Primitive of $\frac 1 {\sin a x + \cos a x}$ ==
=== Primitive of $\dfrac 1 {\sin x + \cos x}$ ===
Category:Examples of Use of Primitive of Reciprocal of Sine of a x plus Cosine of a x
Primitive of Reciprocal of Sine of a x plus Cosine of a x/Examples/sin x + cos x

== Example of Use of Primitive of $\dfrac 1 {\sin a x + \cos a x}$ ==
:$\ds \int \dfrac {\d x} {\sin x + \cos x} = \frac 1 {\sqrt 2} \ln \size {\map \cosec {x + \frac \pi 4} - \map \cot {x + \frac \pi 4} } + C$
== Proof ==
From Primitive of $\dfrac 1 {\sin a x + \cos a x}$:
The result follows on setting $a = 1$.
== Sources ==
* : $\text {II}$. Calculus: Exercises $\text {XIV}$: $27$.
Category:Examples of Use of Primitive of Reciprocal of Sine of a x plus Cosine of a x
Primitive of Reciprocal of Sine of x by Cosine of x

== Theorem ==
:$\ds \int \frac {\d x} {\sin x \cos x} = \ln \size {\tan x} + C$
=== Corollary ===
== Proof ==
== Sources ==
* : Appendix: Table $2$: Integrals
* : Appendix: Table $2$: Integrals
Category:Primitive of Reciprocal of Sine of x by Cosine of x
Category:Primitives involving Sine Function and Cosine Function
Primitive of Reciprocal of Sine of x by Cosine of x/Corollary

== Theorem ==
:$\ds \int \frac {\d x} {\sin a x \cos a x} = \frac 1 a \ln \size {\tan a x} + C$
== Proof ==
== Sources ==
* : $\S 14$: Integrals involving $\sin a x$ and $\cos a x$: $14.404$
Category:Primitive of Reciprocal of Sine of x by Cosine of x
Primitive of Reciprocal of Square of 1 minus Cosine of a x

== Theorem ==
:$\ds \int \frac {\d x} {\paren {1 - \cos a x}^2} = \frac {-1} {2 a} \cot \frac {a x} 2 - \frac 1 {6 a} \cot^3 \frac {a x} 2 + C$
== Proof ==
== Also see ==
* Primitive of $\dfrac 1 {\paren {1 - \sin a x}^2}$
== Sources ==
* : $\S 14$: Integrals involving $\cos a x$: $14.388$
Category:Primitives involving Cosine Function
Primitive of Reciprocal of Square of 1 minus Sine of a x

== Theorem ==
:$\ds \int \frac {\d x} {\paren {1 - \sin a x}^2} = \frac 1 {2 a} \map \tan {\frac \pi 4 + \frac {a x} 2} + \frac 1 {6 a} \map {\tan^3} {\frac \pi 4 + \frac {a x} 2} + C$
== Proof ==
Let:
Thus:
== Also see ==
* Primitive of $\dfrac 1 {\paren {1 - \cos a x}^2}$
== Sources ==
* : $\S 14$: Integrals involving $\sin a x$: $14.358$
Category:Primitives involving Sine Function
Primitive of Reciprocal of Square of 1 plus Cosine of a x

== Theorem ==
:$\ds \int \frac {\d x} {\paren {1 + \cos a x}^2} = \frac 1 {2 a} \tan \frac {a x} 2 + \frac 1 {6 a} \tan^3 \frac {a x} 2 + C$
== Proof ==
== Also see ==
* Primitive of $\dfrac 1 {\paren {1 + \sin a x}^2}$
== Sources ==
* : $\S 14$: Integrals involving $\cos a x$: $14.389$
Category:Primitives involving Cosine Function
Primitive of Reciprocal of Square of 1 plus Sine of a x

== Theorem ==
:$\ds \int \frac {\d x} {\paren {1 + \sin a x}^2} = \frac {-1} {2 a} \map \tan {\frac \pi 4 - \frac {a x} 2} - \frac 1 {6 a} \map {\tan^3} {\frac \pi 4 - \frac {a x} 2} + C$
== Proof ==
Let:
Thus:
== Also see ==
* Primitive of $\dfrac 1 {\paren {1 + \cos a x}^2}$
== Sources ==
* : $\S 14$: Integrals involving $\sin a x$: $14.359$
Category:Primitives involving Sine Function
Primitive of Reciprocal of Square of Cosine of a x

== Theorem ==
:$\ds \int \frac {\d x} {\cos^2 a x} = \frac {\tan a x} a + C$
=== Corollary ===
== Proof ==
== Also see ==
* Primitive of $\dfrac 1 {\sin^2 a x}$
== Sources ==
* : $\S 14$: Integrals involving $\cos a x$: $14.381$
Category:Primitives involving Cosine Function
Primitive of Reciprocal of Square of Cosine of a x/Corollary

== Corollary to Primitive of $\dfrac 1 {\cos^2 a x}$ ==
:$\ds \int \frac {\d x} {\cos^2 x} = \tan x + C$
== Proof ==
== Sources ==
* : $6$. Integral Calculus: Appendix: Table of Fundamental Standard Integrals
Category:Primitive of Reciprocal of Square of Cosine of a x
Category:Primitives involving Cosine Function
Primitive of Reciprocal of Square of Hyperbolic Cosine of a x

== Theorem ==
:$\ds \int \frac {\d x} {\cosh^2 a x} = \frac {\tanh a x} a + C$
== Proof ==
== Also see ==
* Primitive of $\dfrac 1 {\sinh^2 a x}$
== Sources ==
* : $\S 14$: Integrals involving $\cosh a x$: $14.571$
Category:Primitives involving Hyperbolic Cosine Function
Primitive of Reciprocal of Square of Hyperbolic Cosine of a x minus 1

== Theorem ==
:$\ds \int \frac {\d x} {\paren {\cosh a x - 1}^2} = \frac 1 {2 a} \coth \frac {a x} 2 - \frac 1 {6 a} \coth^3 \frac {a x} 2 + C$
== Proof ==
== Also see ==
* Primitive of $\dfrac 1 {\paren {\cosh a x + 1}^2}$
== Sources ==
* : $\S 14$: Integrals involving $\cosh a x$: $14.580$
Category:Primitives involving Hyperbolic Cosine Function
Primitive of Reciprocal of Square of Hyperbolic Cosine of a x plus 1

== Theorem ==
:$\ds \int \frac {\d x} {\paren {\cosh a x + 1}^2} = \frac 1 {2 a} \tanh \frac {a x} 2 - \frac 1 {6 a} \tanh^3 \frac {a x} 2 + C$
== Proof ==
== Also see ==
* Primitive of $\dfrac 1 {\paren {\cosh a x - 1}^2}$
== Sources ==
* : $\S 14$: Integrals involving $\cosh a x$: $14.579$
Category:Primitives involving Hyperbolic Cosine Function
Primitive of Reciprocal of Square of Hyperbolic Sine of a x

== Theorem ==
:$\ds \int \frac {\d x} {\sinh^2 a x} = -\frac {\coth a x} a + C$
== Proof ==
== Also see ==
* Primitive of $\dfrac 1 {\cosh^2 a x}$
== Sources ==
* : $\S 14$: Integrals involving $\sinh a x$: $14.549$
Category:Primitives involving Hyperbolic Sine Function