List of integrals of inverse trigonometric functions

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Categories: Integrals | Mathematics-related lists

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 the Table of Integrals and the List 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.

1 Arcsine
2 Arccosine
3 Arctangent
4 Arccosecant
5 Arcsecant
6 Arccotangent

Arcsine

Failed to parse (Missing texvc executable; please see math/README to configure.): \int \arcsin x \,dx = x\arcsin x+ \sqrt{1-x^2}

Failed to parse (Missing texvc executable; please see math/README to configure.): \int \arcsin \frac{x}{c} \ dx = x \arcsin \frac{x}{c} + \sqrt{c^2 - x^2}

Failed to parse (Missing texvc executable; please see math/README to configure.): \int x \arcsin \frac{x}{c} \ dx = \left( \frac{x^2}{2} - \frac{c^2}{4} \right) \arcsin \frac{x}{c} + \frac{x}{4} \sqrt{c^2 - x^2}

Failed to parse (Missing texvc executable; please see math/README to configure.): \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}

Failed to parse (Missing texvc executable; please see math/README to configure.): \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 \right)

Arccosine

Failed to parse (Missing texvc executable; please see math/README to configure.): \int \arccos x \,dx = x\arccos x- \sqrt{1-x^2}

Failed to parse (Missing texvc executable; please see math/README to configure.): \int \arccos \frac{x}{c} \ dx = x \arccos \frac{x}{c} - \sqrt{c^2 - x^2}

Failed to parse (Missing texvc executable; please see math/README to configure.): \int x \arccos \frac{x}{c} \ dx = \left( \frac{x^2}{2} - \frac{c^2}{4} \right) \arccos \frac{x}{c} - \frac{x}{4} \sqrt{c^2 - x^2}

Failed to parse (Missing texvc executable; please see math/README to configure.): \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

Failed to parse (Missing texvc executable; please see math/README to configure.): \int \arctan x \,dx = x\arctan x- \frac{1}{2}\ln|1+x^2|

Failed to parse (Missing texvc executable; please see math/README to configure.): \int \arctan\big( \frac{x}{c}\big) dx = x \arctan \big( \frac{x}{c} \big) - \frac{c}{2} \ln(1 + \frac{x^2}{c^2})

Failed to parse (Missing texvc executable; please see math/README to configure.): \int x \arctan\big( \frac{x}{c}\big) dx = \frac{ (c^2 + x^2) \arctan \big( \frac{x}{c} \big) - c x}{2}

Failed to parse (Missing texvc executable; please see math/README to configure.): \int x^2 \arctan\big( \frac{x}{c}\big) dx = \frac{x^3}{3} \arctan \big(\frac{x}{c}\big) - \frac{c x^2}{6} + \frac{c^3}{6} \ln|{c^2 + x^2}|

Failed to parse (Missing texvc executable; please see math/README to configure.): \int x^n \arctan \big( \frac{x}{c}\big) dx = \frac{x^{n + 1}}{n + 1} \arctan \big( \frac{x}{c} \big) - \frac{c}{n + 1} \int \frac{x^{n + 1}}{c^2 + x^2} \ dx, \quad n \neq -1

Arccosecant

Failed to parse (Missing texvc executable; please see math/README to configure.): \int \arccsc x \,dx = x\arccsc x+ \ln\left| x+x\sqrt{{x^2-1}\over x^2}\right|

Failed to parse (Missing texvc executable; please see math/README to configure.): \int \arccsc \frac{x}{c} \ dx = x \arccsc \frac{x}{c} + {c} \ln{(\frac{x}{c}(\sqrt{1-\frac{c^2}{x^2}} + 1))}

Failed to parse (Missing texvc executable; please see math/README to configure.): \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

Failed to parse (Missing texvc executable; please see math/README to configure.): \int \arcsec x \,dx = x\arcsec x- \ln\left| x+x\sqrt{{x^2-1}\over x^2}\right|

Failed to parse (Missing texvc executable; please see math/README to configure.): \int \arcsec \frac{x}{c} \ dx = x \arcsec \frac{x}{c} + \frac{x}{c |x|} \ln \left| x \pm \sqrt{x^2 - 1} \right|

Failed to parse (Missing texvc executable; please see math/README to configure.): \int x \arcsec x \ dx = \frac{1}{2} \left( x^2 \arcsec x - \sqrt{x^2 - 1} \right)

Failed to parse (Missing texvc executable; please see math/README to configure.): \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 \right) \right] \right)

Arccotangent

Failed to parse (Missing texvc executable; please see math/README to configure.): \int \arccot x \,dx = x\arccot x+ \frac{1}{2} \ln|1+x^2|

Failed to parse (Missing texvc executable; please see math/README to configure.): \int \arccot \frac{x}{c} \ dx = x \arccot \frac{x}{c} + \frac{c}{2} \ln(c^2 + x^2)

Failed to parse (Missing texvc executable; please see math/README to configure.): \int x \arccot \frac{x}{c} \ dx = \frac{c^2 + x^2}{2} \arccot \frac{x}{c} + \frac{c x}{2}

Failed to parse (Missing texvc executable; please see math/README to configure.): \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)

Failed to parse (Missing texvc executable; please see math/README to configure.): \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 \neq 1

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