What is the integral of sqrt((1-cos^2(x))^3) ?

The integral of sqrt((1-cos^2(x))^3) is -cos(x)+1/3 cos^3(x)+C

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integral of sqrt((1-cos^2(x))^3)

\int (1 - cos^{2} (x))^{3} d x

- cos (x) + \frac{1}{3} cos^{3} (x) + C

Solution steps

\int (1 - cos^{2} (x))^{3} d x

Apply Integration By Parts

: x sin^{4} (x) sin^{2} (x) - \int \frac{3 x sin ^{5} ( x ) cos ( x )}{sin ^{4} ( x ) sin ^{2} ( x )} d x

= x sin^{4} (x) sin^{2} (x) - \int \frac{3 x sin ^{5} ( x ) cos ( x )}{sin ^{4} ( x ) sin ^{2} ( x )} d x

\int \frac{3 x sin ^{5} ( x ) cos ( x )}{sin ^{4} ( x ) sin ^{2} ( x )} d x = x sin^{3} (x) + cos (x) - \frac{1}{3} cos^{3} (x)

= x sin^{4} (x) sin^{2} (x) - (x sin^{3} (x) + cos (x) - \frac{1}{3} cos^{3} (x))

Simplify

x sin^{4} (x) sin^{2} (x) - (x sin^{3} (x) + cos (x) - \frac{1}{3} cos^{3} (x)) : - cos (x) + \frac{1}{3} cos^{3} (x)

= - cos (x) + \frac{1}{3} cos^{3} (x)

Add a constant to the solution

= - cos (x) + \frac{1}{3} cos^{3} (x) + C

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