What is the integral of (arccos(x))/(x^2) ?

The integral of (arccos(x))/(x^2) is -(arccos(x))/x+1/2 ln|sqrt(1-x^2)+1|-1/2 ln|sqrt(1-x^2)-1|+C

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integral of (arccos(x))/(x^2)

\int \frac{arccos ( x )}{x ^{2}} d x

- \frac{arccos ( x )}{x} + \frac{1}{2} ln 1 - x^{2} + 1 - \frac{1}{2} ln 1 - x^{2} - 1 + C

Solution steps

\int \frac{arccos ( x )}{x ^{2}} d x

Apply Integration By Parts

: - \frac{arccos ( x )}{x} - \int \frac{1}{x 1 - x ^{2}} d x

= - \frac{arccos ( x )}{x} - \int \frac{1}{x 1 - x ^{2}} d x

\int \frac{1}{x 1 - x ^{2}} d x = - \frac{1}{2} ln 1 - x^{2} + 1 + \frac{1}{2} ln 1 - x^{2} - 1

= - \frac{arccos ( x )}{x} - (- \frac{1}{2} ln 1 - x^{2} + 1 + \frac{1}{2} ln 1 - x^{2} - 1)

Simplify

= - \frac{arccos ( x )}{x} + \frac{1}{2} ln 1 - x^{2} + 1 - \frac{1}{2} ln 1 - x^{2} - 1

Add a constant to the solution

= - \frac{arccos ( x )}{x} + \frac{1}{2} ln 1 - x^{2} + 1 - \frac{1}{2} ln 1 - x^{2} - 1 + C

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What is the integral of (arccos(x))/(x^2) ?
The integral of (arccos(x))/(x^2) is -(arccos(x))/x+1/2 ln|sqrt(1-x^2)+1|-1/2 ln|sqrt(1-x^2)-1|+C