# List of integrals of inverse trigonometric functions

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List of integrals of inverse trigonometric functions

The following is a list of integrals (antiderivative formulas) for integrands that contain inverse trigonometric functions (also known as "arc functions"). For a complete list of integral formulas, see lists of integrals.

Note: There are three common notations for inverse trigonometric functions. The arcsine function, for instance, could be written as "sin−1", "asin", or, as is used on this page, "arcsin".

Arcsine

:$int arcsin x ,dx = xarcsin x+ sqrt\left\{1-x^2\right\}$

:$int arcsin frac\left\{x\right\}\left\{c\right\} dx = x arcsin frac\left\{x\right\}\left\{c\right\} + sqrt\left\{c^2 - x^2\right\}$

:$int x arcsin frac\left\{x\right\}\left\{c\right\} dx = left\left( frac\left\{x^2\right\}\left\{2\right\} - frac\left\{c^2\right\}\left\{4\right\} ight\right) arcsin frac\left\{x\right\}\left\{c\right\} + frac\left\{x\right\}\left\{4\right\} sqrt\left\{c^2 - x^2\right\}$

:$int x^2 arcsin frac\left\{x\right\}\left\{c\right\} dx = frac\left\{x^3\right\}\left\{3\right\} arcsin frac\left\{x\right\}\left\{c\right\} + frac\left\{x^2 + 2c^2\right\}\left\{9\right\} sqrt\left\{c^2 - x^2\right\}$

:$int x^n arcsin x dx = frac\left\{1\right\}\left\{n + 1\right\} left\left( x^\left\{n + 1\right\} arcsin x + frac\left\{x^n sqrt\left\{1 - x^2\right\} - n x^\left\{n - 1\right\} arcsin x\right\}\left\{n - 1\right\} + n int x^\left\{n - 2\right\} arcsin x dx ight\right)$

Arccosine

:$int arccos x ,dx = xarccos x- sqrt\left\{1-x^2\right\}$

:$int arccos frac\left\{x\right\}\left\{c\right\} dx = x arccos frac\left\{x\right\}\left\{c\right\} - sqrt\left\{c^2 - x^2\right\}$

:$int x arccos frac\left\{x\right\}\left\{c\right\} dx = left\left( frac\left\{x^2\right\}\left\{2\right\} - frac\left\{c^2\right\}\left\{4\right\} ight\right) arccos frac\left\{x\right\}\left\{c\right\} - frac\left\{x\right\}\left\{4\right\} sqrt\left\{c^2 - x^2\right\}$

:$int x^2 arccos frac\left\{x\right\}\left\{c\right\} dx = frac\left\{x^3\right\}\left\{3\right\} arccos frac\left\{x\right\}\left\{c\right\} - frac\left\{x^2 + 2c^2\right\}\left\{9\right\} sqrt\left\{c^2 - x^2\right\}$

Arctangent

:$int arctan x ,dx = xarctan x- frac\left\{1\right\}\left\{2\right\}ln|1+x^2|$

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Arccosecant

:$int arccsc x ,dx = xarccsc x+ lnleft| x+xsqrt$x^2-1}over x^2} ight|

:$int arccsc frac\left\{x\right\}\left\{c\right\} dx = x arccsc frac\left\{x\right\}\left\{c\right\} + \left\{c\right\} ln\left\{\left(frac\left\{x\right\}\left\{c\right\}\left(sqrt\left\{1-frac\left\{c^2\right\}\left\{x^2$ + 1))}

:$int x arccsc frac\left\{x\right\}\left\{c\right\} dx = frac\left\{x^2\right\}\left\{2\right\} arccsc frac\left\{x\right\}\left\{c\right\} + frac\left\{cx\right\}\left\{2\right\} sqrt\left\{1-frac\left\{c^2\right\}\left\{x^2$

Arcsecant

:$int arcsec x ,dx = xarcsec x- lnleft| x+xsqrt$x^2-1}over x^2} ight|

:$int arcsec frac\left\{x\right\}\left\{c\right\} dx = x arcsec frac\left\{x\right\}\left\{c\right\} + frac\left\{x\right\}\left\{c |x ln left| x pm sqrt\left\{x^2 - 1\right\} ight|$

:$int x arcsec x dx = frac\left\{1\right\}\left\{2\right\} left\left( x^2 arcsec x - sqrt\left\{x^2 - 1\right\} ight\right)$

:$int x^n arcsec x dx = frac\left\{1\right\}\left\{n + 1\right\} left\left( x^\left\{n + 1\right\} arcsec x - frac\left\{1\right\}\left\{n\right\} left \left[ x^\left\{n - 1\right\} sqrt\left\{x^2 - 1\right\} + \left(1 - n\right) left\left( x^\left\{n - 1\right\} arcsec x + \left(1 - n\right) int x^\left\{n - 2\right\} arcsec x dx ight\right) ight\right] ight\right)$

Arccotangent

:$int arccot x ,dx = xarccot x+ frac\left\{1\right\}\left\{2\right\} ln|1+x^2|$

:$int arccot frac\left\{x\right\}\left\{c\right\} dx = x arccot frac\left\{x\right\}\left\{c\right\} + frac\left\{c\right\}\left\{2\right\} ln\left(c^2 + x^2\right)$

:$int x arccot frac\left\{x\right\}\left\{c\right\} dx = frac\left\{c^2 + x^2\right\}\left\{2\right\} arccot frac\left\{x\right\}\left\{c\right\} + frac\left\{c x\right\}\left\{2\right\}$

:$int x^2 arccot frac\left\{x\right\}\left\{c\right\} dx = frac\left\{x^3\right\}\left\{3\right\} arccot frac\left\{x\right\}\left\{c\right\} + frac\left\{c x^2\right\}\left\{6\right\} - frac\left\{c^3\right\}\left\{6\right\} ln\left(c^2 + x^2\right)$

:$int x^n arccot frac\left\{x\right\}\left\{c\right\} dx = frac\left\{x^\left\{n + 1${n+1} arccot frac{x}{c} + frac{c}{n + 1} int frac{x^{n + 1{c^2 + x^2} dx, quad n eq 1

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