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List of integrals of inverse trigonometric functions

 
Wikipedia: List of integrals of inverse trigonometric functions
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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.

C is used for the arbitrary constant of integration that can only be determined if something about the value of the integral at some point is known. Thus each function has an infinite number of antiderivatives.

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.

Contents

Arcsine

\int \arcsin x \,dx = x\arcsin x+ \sqrt{1-x^2} + C
\int \arcsin \frac{x}{a} \ dx = x \arcsin \frac{x}{a} + \sqrt{a^2 - x^2} + C
\int x \arcsin \frac{x}{a} \ dx = \left( \frac{x^2}{2} - \frac{a^2}{4} \right) \arcsin \frac{x}{a} + \frac{x}{4} \sqrt{a^2 - x^2} + C
\int x^2 \arcsin \frac{x}{a} \ dx = \frac{x^3}{3} \arcsin \frac{x}{a} + \frac{x^2 + 2a^2}{9} \sqrt{a^2 - x^2} + C
\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)
\int \cos^n x \arcsin x \ dx = \left( x^{n^2 + 1} \arccos x + \frac{x^n \sqrt{1 - x^4} - n x^{n^2 - 1} \arccos x}{n^2 - 1} + n \int x^{n^2 - 2} \arccos x \ dx \right)

Arccosine

\int \arccos x \,dx = x\arccos x- \sqrt{1-x^2} + C
\int \arccos \frac{x}{a} \ dx = x \arccos \frac{x}{a} - \sqrt{a^2 - x^2} + C
\int x \arccos \frac{x}{a} \ dx = \left( \frac{x^2}{2} - \frac{a^2}{4} \right) \arccos \frac{x}{a} - \frac{x}{4} \sqrt{a^2 - x^2} + C
\int x^2 \arccos \frac{x}{a} \ dx = \frac{x^3}{3} \arccos \frac{x}{a} - \frac{x^2 + 2a^2}{9} \sqrt{a^2 - x^2} + C

Arctangent

\int \arctan x \,dx = x\arctan x- \frac{1}{2}\ln(1+x^2) + C
\int \arctan\big( \frac{x}{a}\big) dx = x \arctan \big( \frac{x}{a} \big) - \frac{a}{2} \ln(1 + \frac{x^2}{a^2}) + C
\int x \arctan\big( \frac{x}{a}\big) dx = \frac{ (a^2 + x^2) \arctan \big( \frac{x}{a} \big) - a x}{2} + C
\int x^2 \arctan\big( \frac{x}{a}\big) dx = \frac{x^3}{3} \arctan \big(\frac{x}{a}\big) - \frac{a x^2}{6} + \frac{a^3}{6} \ln({a^2 + x^2}) + C
\int x^n \arctan \big( \frac{x}{a}\big) dx = \frac{x^{n + 1}}{n + 1} \arctan \big( \frac{x}{a} \big) - \frac{a}{n + 1} \int \frac{x^{n + 1}}{a^2 + x^2} \ dx, \quad n \neq -1

Arccosecant

\int \arccsc x \,dx = x\arccsc x+ \ln\left| x+x\sqrt{{x^2-1}\over x^2}\right| + C
\int \arccsc \frac{x}{a} \ dx = x \arccsc \frac{x}{a} + {a} \ln{(\frac{x}{a}(\sqrt{1-\frac{a^2}{x^2}} + 1))} + C
\int x \arccsc \frac{x}{a} \ dx = \frac{x^2}{2} \arccsc \frac{x}{a} + \frac{ax}{2} \sqrt{1-\frac{a^2}{x^2}} + C

Arcsecant

\int \arcsec x \,dx = x\arcsec x- \ln\left| x+x\sqrt{{x^2-1}\over x^2}\right| + C
\int \arcsec \frac{x}{a} \ dx = x \arcsec \frac{x}{a} + \frac{x}{a |x|} \ln \left| x \pm \sqrt{x^2 - 1} \right| + C
\int x \arcsec x \ dx = \frac{1}{2} \left( x^2 \arcsec x - \sqrt{x^2 - 1} \right) + C
\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

\int \arccot x \,dx = x\arccot x+ \frac{1}{2} \ln(1+x^2) + C
\int \arccot \frac{x}{a} \ dx = x \arccot \frac{x}{a} + \frac{a}{2} \ln(a^2 + x^2) + C
\int x \arccot \frac{x}{a} \ dx = \frac{a^2 + x^2}{2} \arccot \frac{x}{a} + \frac{a x}{2} + C
\int x^2 \arccot \frac{x}{a} \ dx = \frac{x^3}{3} \arccot \frac{x}{a} + \frac{a x^2}{6} - \frac{a^3}{6} \ln(a^2 + x^2) + C
\int x^n \arccot \frac{x}{a} \ dx = \frac{x^{n + 1}}{n+1} \arccot \frac{x}{a} + \frac{a}{n + 1} \int \frac{x^{n + 1}}{a^2 + x^2} \ dx, \quad n \neq -1


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Lists of integrals

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