List of integrals of arc functions

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The following is a list of integrals (antiderivative formulas) for integrands that contain inverse trigonometric functions (also know 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 trig. 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 Arcsecant
5 Arccotangent
6 Arccosecant

[edit] Arcsine

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

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

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

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

[edit] Arccosine

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

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

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

[edit] Arctangent

$\int \arctan\bigg( \frac{x}{c}\bigg) dx = x \arctan \bigg( \frac{x}{c} \bigg) - \frac{c}{2} \ln(c^2 + x^2)$

$\int x \arctan\bigg( \frac{x}{c}\bigg) dx = \frac{ (c^2 + x^2) \arctan \bigg( \frac{x}{c} \bigg) - c x}{2}$

$\int x^2 \arctan\bigg( \frac{x}{c}\bigg) dx = \frac{x^3}{3} \arctan \bigg(\frac{x}{c}\bigg) - \frac{c x^2}{6} + \frac{c^3}{6} \ln{c^2 + x^2}$

$\int x^n \arctan \bigg( \frac{x}{c}\bigg) dx = \frac{x^{n + 1}}{n + 1} \arctan \bigg( \frac{x}{c} \bigg) - \frac{c}{n + 1} \int \frac{x^{n + 1}}{c^2 + x^2} \ dx, \quad n \neq 1$

[edit] Arcsecant

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

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

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

[edit] Arccotangent

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

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

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

$\int x^n \arccot \frac{x}{c} \ dx = \frac{x^{n + 1}}{n+1} \arccot \frac{x}{c} + \frac{c}{n + 1} \int \frac{x^{n + 1}}{c^2 + x^2} \ dx, \quad n \neq 1$