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.

Arcsine

• <math>\int \arcsin \frac{x}{c} \ dx = x \arcsin \frac{x}{c} + \sqrt{c^2 - x^2}</math>

• <math>\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}</math>

• <math>\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}</math>

• <math>\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)</math>

Arccosine

• <math>\int \arccos \frac{x}{c} \ dx = x \arccos \frac{x}{c} - \sqrt{c^2 - x^2}</math>

• <math>\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}</math>

• <math>\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}</math>

Arctangent

• <math>\int \arctan \frac{x}{c} \ dx = x \arctan \frac{x}{c} - \frac{c}{2} \ln(c^2 + x^2)</math>

• <math>\int x \arctan \frac{x}{c} \ dx = \frac{c^2 + x^2}{2} \arctan \frac{x}{c} - \frac{c x}{2}</math>

• <math>\int x^2 \arctan \frac{x}{c} \ dx = \frac{x^3}{3} \arctan \frac{x}{c} - \frac{c x^2}{6} + \frac{c^3}{6} \ln{c^2 + x^2}</math>

• <math>\int x^n \arctan \frac{x}{c} \ dx = \frac{x^{n + 1}}{n + 1} \arctan \frac{x}{c} - \frac{c}{n + 1} \int \frac{x^{n + 1}}{c^2 + x^2} \ dx, \quad n \neq 1</math>

Arcsecant

• <math>\int \arcsec \frac{x}{c} \ dx = x \arcsec \frac{x}{c} + \frac{x}{c |x|} \ln \left| x \pm \sqrt{x^2 - 1} \right|</math>

• <math>\int x \arcsec x \ dx = \frac{1}{2} \left( x^2 \arcsec x - \sqrt{x^2 - 1} \right)</math>

• <math>\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)</math>

Arccotangent

• <math>\int \arccot \frac{x}{c} \ dx = x \arccot \frac{x}{c} + \frac{c}{2} \ln(c^2 + x^2)</math>

• <math>\int x \arccot \frac{x}{c} \ dx = \frac{c^2 + x^2}{2} \arccot \frac{x}{c} + \frac{c x}{2}</math>

• <math>\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)</math>

• <math>\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</math>es:Lista de integrales de funciones inversas trigonométricas

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