View source for List of integrals of inverse trigonometric functions

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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 ==

{{startplainlist|indent=1}}
* <math>\int\arcsin(x)\,dx=
  x\arcsin(x)+
 {\sqrt{1-x^2}}+C</math>
* <math>\int\arcsin(ax)\,dx= x\arcsin(ax)+
  \frac{\sqrt{1-a^2x^2}}{a}+C</math>
* <math>\int x\arcsin(ax)\,dx=
  \frac{x^2\arcsin(ax)}{2}-
  \frac{\arcsin(ax)}{4\,a^2}+
  \frac{x\sqrt{1-a^2x^2}}{4\,a}+C</math>
* <math>\int x^2\arcsin(ax)\,dx=
  \frac{x^3\arcsin(ax)}{3}+
  \frac{\left(a^2x^2+2\right)\sqrt{1-a^2x^2}}{9\,a^3}+C</math>
* <math>\int x^m\arcsin(ax)\,dx=
  \frac{x^{m+1}\arcsin(ax)}{m+1}\,-\,
  \frac{a}{m+1}\int \frac{x^{m+1}}{\sqrt{1-a^2x^2}}\,dx\,,\quad(m\ne-1)</math>
* <math>\int\arcsin(ax)^2\,dx=
  -2x+x\arcsin(ax)^2+
  \frac{2\sqrt{1-a^2x^2}\arcsin(ax)}{a}+C</math>
* <math>\int\arcsin(ax)^n\,dx=
  x\arcsin(ax)^n\,+\,
  \frac{n\sqrt{1-a^2x^2}\arcsin(ax)^{n-1}}{a}\,-\,
  n\,(n-1)\int\arcsin(ax)^{n-2}\,dx</math>
* <math>\int\arcsin(ax)^n\,dx=
  \frac{x\arcsin(ax)^{n+2}}{(n+1)\,(n+2)}\,+\,
  \frac{\sqrt{1-a^2x^2}\arcsin(ax)^{n+1}}{a\,(n+1)}\,-\,
  \frac{1}{(n+1)\,(n+2)}\int\arcsin(ax)^{n+2}\,dx\,,\quad(n\ne-1,-2)</math>
{{endplainlist}}

== Arccosine function integration formulas ==

{{startplainlist|indent=1}}
* <math>\int\arccos(x)\,dx=
  x\arccos(x)-
  {\sqrt{1-x^2}}+C</math>
* <math>\int\arccos(ax)\,dx=
  x\arccos(ax)-
  \frac{\sqrt{1-a^2x^2}}{a}+C</math>
* <math>\int x\arccos(ax)\,dx=
  \frac{x^2\arccos(ax)}{2}-
  \frac{\arccos(ax)}{4\,a^2}-
  \frac{x\sqrt{1-a^2x^2}}{4\,a}+C</math>
* <math>\int x^2\arccos(ax)\,dx=
  \frac{x^3\arccos(ax)}{3}-
  \frac{\left(a^2x^2+2\right)\sqrt{1-a^2x^2}}{9\,a^3}+C</math>
* <math>\int x^m\arccos(ax)\,dx=
  \frac{x^{m+1}\arccos(ax)}{m+1}\,+\,
  \frac{a}{m+1}\int \frac{x^{m+1}}{\sqrt{1-a^2x^2}}\,dx\,,\quad(m\ne-1)</math>
* <math>\int\arccos(ax)^2\,dx=
  -2x+x\arccos(ax)^2-
  \frac{2\sqrt{1-a^2x^2}\arccos(ax)}{a}+C</math>
* <math>\int\arccos(ax)^n\,dx=
  x\arccos(ax)^n\,-\,
  \frac{n\sqrt{1-a^2x^2}\arccos(ax)^{n-1}}{a}\,-\,
  n\,(n-1)\int\arccos(ax)^{n-2}\,dx</math>
* <math>\int\arccos(ax)^n\,dx=
  \frac{x\arccos(ax)^{n+2}}{(n+1)\,(n+2)}\,-\,
  \frac{\sqrt{1-a^2x^2}\arccos(ax)^{n+1}}{a\,(n+1)}\,-\,
  \frac{1}{(n+1)\,(n+2)}\int\arccos(ax)^{n+2}\,dx\,,\quad(n\ne-1,-2)</math>
{{endplainlist}}

== Arctangent function integration formulas ==

{{startplainlist|indent=1}}
* <math>\int\arctan(x)\,dx=
  x\arctan(x)-
  \frac{\ln\left(x^2+1\right)}{2}+C</math>
* <math>\int\arctan(ax)\,dx=
  x\arctan(ax)-
  \frac{\ln\left(a^2x^2+1\right)}{2\,a}+C</math>
* <math>\int x\arctan(ax)\,dx=
  \frac{x^2\arctan(ax)}{2}+
  \frac{\arctan(ax)}{2\,a^2}-\frac{x}{2\,a}+C</math>
* <math>\int x^2\arctan(ax)\,dx=
  \frac{x^3\arctan(ax)}{3}+
  \frac{\ln\left(a^2x^2+1\right)}{6\,a^3}-\frac{x^2}{6\,a}+C</math>
* <math>\int x^m\arctan(ax)\,dx=
  \frac{x^{m+1}\arctan(ax)}{m+1}-
  \frac{a}{m+1}\int \frac{x^{m+1}}{a^2x^2+1}\,dx\,,\quad(m\ne-1)</math>
{{endplainlist}}

== Arccotangent function integration formulas ==

{{startplainlist|indent=1}}
* <math>\int\arccot(x)\,dx=
  x\arccot(x)+
  \frac{\ln\left(x^2+1\right)}{2}+C</math>
* <math>\int\arccot(ax)\,dx=
  x\arccot(ax)+
  \frac{\ln\left(a^2x^2+1\right)}{2\,a}+C</math>
* <math>\int x\arccot(ax)\,dx=
  \frac{x^2\arccot(ax)}{2}+
  \frac{\arccot(ax)}{2\,a^2}+\frac{x}{2\,a}+C</math>
* <math>\int x^2\arccot(ax)\,dx=
  \frac{x^3\arccot(ax)}{3}-
  \frac{\ln\left(a^2x^2+1\right)}{6\,a^3}+\frac{x^2}{6\,a}+C</math>
* <math>\int x^m\arccot(ax)\,dx=
  \frac{x^{m+1}\arccot(ax)}{m+1}+
  \frac{a}{m+1}\int \frac{x^{m+1}}{a^2x^2+1}\,dx\,,\quad(m\ne-1)</math>
{{endplainlist}}

== Arcsecant function integration formulas ==

{{startplainlist|indent=1}}
* <math>\int\arcsec(x)\,dx= x\arcsec(x) \, - \,
 \ln\left(\left|x\right|+\sqrt{x^2-1}\right)\,+\,C=
  x\arcsec(x)-\operatorname{arcosh}|x|+C</math>
* <math>\int\arcsec(ax)\,dx=
  x\arcsec(ax)-
  \frac{1}{a}\,\operatorname{arcosh}|ax|+C</math>
* <math>\int x\arcsec(ax)\,dx=
  \frac{x^2\arcsec(ax)}{2}-
  \frac{x}{2\,a}\sqrt{1-\frac{1}{a^2x^2}}+C</math>
* <math>\int x^2\arcsec(ax)\,dx=
  \frac{x^3\arcsec(ax)}{3}\,-\,
  \frac{\operatorname{arcosh}|ax|}{6\,a^3}\,-\,
  \frac{x^2}{6\,a}\sqrt{1-\frac{1}{a^2x^2}}\,+\,C</math>
* <math>\int x^m\arcsec(ax)\,dx=
  \frac{x^{m+1}\arcsec(ax)}{m+1}\,-\,
  \frac{1}{a\,(m+1)}\int \frac{x^{m-1}}{\sqrt{1-\frac{1}{a^2x^2}}}\,dx\,,\quad(m\ne-1)</math>
{{endplainlist}}

== Arccosecant function integration formulas ==

{{startplainlist|indent=1}}
* <math>\int\arccsc(x)\,dx= 
  x\arccsc(x) \, + \, \ln\left(\left|x\right|+\sqrt{x^2-1}\right)\,+\,C=
  x\arccsc(x)\,+\, \operatorname{arcosh}|x|\,+\,C </math>
* <math>\int\arccsc(ax)\,dx= x\arccsc(ax)+
  \frac{1}{a}\,\operatorname{artanh}\,\sqrt{1-\frac{1}{a^2x^2}}+C</math>
* <math>\int x\arccsc(ax)\,dx=
  \frac{x^2\arccsc(ax)}{2}+
  \frac{x}{2\,a}\sqrt{1-\frac{1}{a^2x^2}}+C</math>
* <math>\int x^2\arccsc(ax)\,dx=
  \frac{x^3\arccsc(ax)}{3}\,+\,
  \frac{1}{6\,a^3}\,\operatorname{artanh}\,\sqrt{1-\frac{1}{a^2x^2}}\,+\,
  \frac{x^2}{6\,a}\sqrt{1-\frac{1}{a^2x^2}}\,+\,C</math>
* <math>\int x^m\arccsc(ax)\,dx=
  \frac{x^{m+1}\arccsc(ax)}{m+1}\,+\,
  \frac{1}{a\,(m+1)}\int \frac{x^{m-1}}{\sqrt{1-\frac{1}{a^2x^2}}}\,dx\,,\quad(m\ne-1)</math>
{{endplainlist}}

==See also==

* {{annotated link|Differentiation of trigonometric functions}}
* {{annotated link|List of trigonometric identities}}
* {{annotated link|Lists of integrals}}

==References==

{{reflist}}

{{Lists of integrals}}

[[Category:Lists of integrals|Arc functions]]