List of integrals of inverse trigonometric functions


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{1-x^2}

:int arcsin frac{x}{c} dx = x arcsin frac{x}{c} + sqrt{c^2 - x^2}

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

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

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

Arccosine

:int arccos x ,dx = xarccos x- sqrt{1-x^2}

:int arccos frac{x}{c} dx = x arccos frac{x}{c} - sqrt{c^2 - x^2}

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

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

Arctangent

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

:int arctanig( frac{x}{c}ig) dx = x arctan ig( frac{x}{c} ig) - frac{c}{2} ln(1 + frac{x^2}{c^2})

:int x arctanig( frac{x}{c}ig) dx = frac{ (c^2 + x^2) arctan ig( frac{x}{c} ig) - c x}{2}

:int x^2 arctanig( frac{x}{c}ig) dx = frac{x^3}{3} arctan ig(frac{x}{c}ig) - frac{c x^2}{6} + frac{c^3}{6} ln|{c^2 + x^2}|

:int x^n arctan ig( frac{x}{c}ig) dx = frac{x^{n + 1{n + 1} arctan ig( frac{x}{c} ig) - frac{c}{n + 1} int frac{x^{n + 1{c^2 + x^2} dx, quad n eq -1

Arccosecant

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

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

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

Arcsecant

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

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

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

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

Arccotangent

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

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

:int x arccot frac{x}{c} dx = frac{c^2 + x^2}{2} arccot frac{x}{c} + frac{c x}{2}

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

:int x^n arccot frac{x}{c} dx = frac{x^{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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