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{{Trigonometry}}
The following is a list of [[indefinite integral]]s ([[antiderivative]]s) of expressions involving the [[inverse trigonometric function]]s. For a complete list of integral formulas, see [[lists of integrals]].

* The inverse trigonometric functions are also known as the "arc functions".
* ''C'' is used for the arbitrary [[constant of integration]] that can only be determined if something about the value of the integral at some point is known. Thus each function has an infinite number of antiderivatives.
* There are three common notations for inverse trigonometric functions. The arcsine function, for instance, could be written as ''sin<sup>−1</sup>'', ''asin'', or, as is used on this page, ''arcsin''.
* For each inverse trigonometric integration formula below there is a corresponding formula in the [[list of integrals of inverse hyperbolic functions]].

== Arcsine function integration formulas ==

:<math>\int\arcsin(x)\,dx=
x\arcsin(x)+
{\sqrt{1-x^2}}+C</math>

:<math>\int x\arcsin(a\,x)\,dx=
\frac{x^2\arcsin(a\,x)}{2}-
\frac{\arcsin(a\,x)}{4\,a^2}+
\frac{x\sqrt{1-a^2\,x^2}}{4\,a}+C</math>

:<math>\int x^2\arcsin(a\,x)\,dx=
\frac{x^3\arcsin(a\,x)}{3}+
\frac{\left(a^2\,x^2+2\right)\sqrt{1-a^2\,x^2}}{9\,a^3}+C</math>

:<math>\int x^m\arcsin(a\,x)\,dx=
\frac{x^{m+1}\arcsin(a\,x)}{m+1}\,-\,
\frac{a}{m+1}\int \frac{x^{m+1}}{\sqrt{1-a^2\,x^2}}\,dx\quad(m\ne-1)</math>

:<math>\int\arcsin(a\,x)^2\,dx=
-2\,x+x\arcsin(a\,x)^2+
\frac{2\sqrt{1-a^2\,x^2}\arcsin(a\,x)}{a}+C</math>

:<math>\int\arcsin(a\,x)^n\,dx=
x\arcsin(a\,x)^n\,+\,
\frac{n\sqrt{1-a^2\,x^2}\arcsin(a\,x)^{n-1}}{a}\,-\,
n\,(n-1)\int\arcsin(a\,x)^{n-2}\,dx</math>

:<math>\int\arcsin(a\,x)^n\,dx=
\frac{x\arcsin(a\,x)^{n+2}}{(n+1)\,(n+2)}\,+\,
\frac{\sqrt{1-a^2\,x^2}\arcsin(a\,x)^{n+1}}{a\,(n+1)}\,-\,
\frac{1}{(n+1)\,(n+2)}\int\arcsin(a\,x)^{n+2}\,dx\quad(n\ne-1,-2)</math>

== Arccosine function integration formulas ==
:<math>\int\arccos(x)\,dx=
x\arccos(x)-
{\sqrt{1-x^2}}+C</math>

:<math>\int\arccos(a\,x)\,dx=
x\arccos(a\,x)-
\frac{\sqrt{1-a^2\,x^2}}{a}+C</math>

:<math>\int x\arccos(a\,x)\,dx=
\frac{x^2\arccos(a\,x)}{2}-
\frac{\arccos(a\,x)}{4\,a^2}-
\frac{x\sqrt{1-a^2\,x^2}}{4\,a}+C</math>

:<math>\int x^2\arccos(a\,x)\,dx=
\frac{x^3\arccos(a\,x)}{3}-
\frac{\left(a^2\,x^2+2\right)\sqrt{1-a^2\,x^2}}{9\,a^3}+C</math>

:<math>\int x^m\arccos(a\,x)\,dx=
\frac{x^{m+1}\arccos(a\,x)}{m+1}\,+\,
\frac{a}{m+1}\int \frac{x^{m+1}}{\sqrt{1-a^2\,x^2}}\,dx\quad(m\ne-1)</math>

:<math>\int\arccos(a\,x)^2\,dx=
-2\,x+x\arccos(a\,x)^2-
\frac{2\sqrt{1-a^2\,x^2}\arccos(a\,x)}{a}+C</math>

:<math>\int\arccos(a\,x)^n\,dx=
x\arccos(a\,x)^n\,-\,
\frac{n\sqrt{1-a^2\,x^2}\arccos(a\,x)^{n-1}}{a}\,-\,
n\,(n-1)\int\arccos(a\,x)^{n-2}\,dx</math>

:<math>\int\arccos(a\,x)^n\,dx=
\frac{x\arccos(a\,x)^{n+2}}{(n+1)\,(n+2)}\,-\,
\frac{\sqrt{1-a^2\,x^2}\arccos(a\,x)^{n+1}}{a\,(n+1)}\,-\,
\frac{1}{(n+1)\,(n+2)}\int\arccos(a\,x)^{n+2}\,dx\quad(n\ne-1,-2)</math>

== Arctangent function integration formulas ==
:<math>\int\arctan(x)\,dx=
x\arctan(x)-
\frac{\ln\left(x^2+1\right)}{2}+C</math>

:<math>\int\arctan(a\,x)\,dx=
x\arctan(a\,x)-
\frac{\ln\left(a^2\,x^2+1\right)}{2\,a}+C</math>

:<math>\int x\arctan(a\,x)\,dx=
\frac{x^2\arctan(a\,x)}{2}+
\frac{\arctan(a\,x)}{2\,a^2}-\frac{x}{2\,a}+C</math>

:<math>\int x^2\arctan(a\,x)\,dx=
\frac{x^3\arctan(a\,x)}{3}+
\frac{\ln\left(a^2\,x^2+1\right)}{6\,a^3}-\frac{x^2}{6\,a}+C</math>

:<math>\int x^m\arctan(a\,x)\,dx=
\frac{x^{m+1}\arctan(a\,x)}{m+1}-
\frac{a}{m+1}\int \frac{x^{m+1}}{a^2\,x^2+1}\,dx\quad(m\ne-1)</math>

== Arccotangent function integration formulas ==

:<math>\int\arccot(x)\,dx=
x\arccot(x)+
\frac{\ln\left(x^2+1\right)}{2}+C</math>

:<math>\int\arccot(a\,x)\,dx=
x\arccot(a\,x)+
\frac{\ln\left(a^2\,x^2+1\right)}{2\,a}+C</math>

:<math>\int x\arccot(a\,x)\,dx=
\frac{x^2\arccot(a\,x)}{2}+
\frac{\arccot(a\,x)}{2\,a^2}+\frac{x}{2\,a}+C</math>

:<math>\int x^2\arccot(a\,x)\,dx=
\frac{x^3\arccot(a\,x)}{3}-
\frac{\ln\left(a^2\,x^2+1\right)}{6\,a^3}+\frac{x^2}{6\,a}+C</math>

:<math>\int x^m\arccot(a\,x)\,dx=
\frac{x^{m+1}\arccot(a\,x)}{m+1}+
\frac{a}{m+1}\int \frac{x^{m+1}}{a^2\,x^2+1}\,dx\quad(m\ne-1)</math>

== Arcsecant function integration formulas ==

:<math>\int\arcsec(x)\,dx=
x\arcsec(x)-\operatorname{arctan}\,\sqrt{1-\frac{1}{x^2}}+C</math>

:<math>\int\arcsec(a\,x)\,dx=
x\arcsec(a\,x)-
\frac{1}{a}\,\operatorname{artanh}\,\sqrt{1-\frac{1}{a^2\,x^2}}+C</math>

:<math>\int x\arcsec(a\,x)\,dx=
\frac{x^2\arcsec(a\,x)}{2}-
\frac{x}{2\,a}\sqrt{1-\frac{1}{a^2\,x^2}}+C</math>

:<math>\int x^2\arcsec(a\,x)\,dx=
\frac{x^3\arcsec(a\,x)}{3}\,-\,
\frac{1}{6\,a^3}\,\operatorname{artanh}\,\sqrt{1-\frac{1}{a^2\,x^2}}\,-\,
\frac{x^2}{6\,a}\sqrt{1-\frac{1}{a^2\,x^2}}\,+\,C</math>

:<math>\int x^m\arcsec(a\,x)\,dx=
\frac{x^{m+1}\arcsec(a\,x)}{m+1}\,-\,
\frac{1}{a\,(m+1)}\int \frac{x^{m-1}}{\sqrt{1-\frac{1}{a^2\,x^2}}}\,dx\quad(m\ne-1)</math>

== Arccosecant function integration formulas ==
:<math>\int\arccsc(x)\,dx=
x\arccos(x)-
\sqrt{1-{x^2}}+C</math>

:<math>\int\arccsc(a\,x)\,dx=
x\arccsc(a\,x)+
\frac{1}{a}\,\operatorname{artanh}\,\sqrt{1-\frac{1}{a^2\,x^2}}+C</math>

:<math>\int x\arccsc(a\,x)\,dx=
\frac{x^2\arccsc(a\,x)}{2}+
\frac{x}{2\,a}\sqrt{1-\frac{1}{a^2\,x^2}}+C</math>

:<math>\int x^2\arccsc(a\,x)\,dx=
\frac{x^3\arccsc(a\,x)}{3}\,+\,
\frac{1}{6\,a^3}\,\operatorname{artanh}\,\sqrt{1-\frac{1}{a^2\,x^2}}\,+\,
\frac{x^2}{6\,a}\sqrt{1-\frac{1}{a^2\,x^2}}\,+\,C</math>

:<math>\int x^m\arccsc(a\,x)\,dx=
\frac{x^{m+1}\arccsc(a\,x)}{m+1}\,+\,
\frac{1}{a\,(m+1)}\int \frac{x^{m-1}}{\sqrt{1-\frac{1}{a^2\,x^2}}}\,dx\quad(m\ne-1)</math>

{{Lists of integrals}}

[[Category:Integrals|Arc functions]]
[[Category:Mathematics-related lists|Integrals of arc functions]]

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