What is the integral of (arccos(x))^3 ?

The integral of (arccos(x))^3 is xarccos^3(x)-3(arccos^2(x)sqrt(1-x^2)-2(-xarccos(x)+sqrt(1-x^2)))+C

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integral of (arccos(x))^3

\int (arccos (x))^{3} d x

x arccos^{3} (x) - 3 (arccos^{2} (x) 1 - x^{2} - 2 (- x arccos (x) + 1 - x^{2})) + C

Solution steps

\int (arccos (x))^{3} d x

Simplify

= \int arccos^{3} (x) d x

Apply Integration By Parts

: x arccos^{3} (x) - \int - \frac{3 x arccos ^{2} ( x )}{1 - x ^{2}} d x

= x arccos^{3} (x) - \int - \frac{3 x arccos ^{2} ( x )}{1 - x ^{2}} d x

\int - \frac{3 x arccos ^{2} ( x )}{1 - x ^{2}} d x = 3 (arccos^{2} (x) 1 - x^{2} - 2 (- x arccos (x) + 1 - x^{2}))

= x arccos^{3} (x) - 3 (arccos^{2} (x) 1 - x^{2} - 2 (- x arccos (x) + 1 - x^{2}))

Add a constant to the solution

= x arccos^{3} (x) - 3 (arccos^{2} (x) 1 - x^{2} - 2 (- x arccos (x) + 1 - x^{2})) + C

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What is the integral of (arccos(x))^3 ?
The integral of (arccos(x))^3 is xarccos^3(x)-3(arccos^2(x)sqrt(1-x^2)-2(-xarccos(x)+sqrt(1-x^2)))+C