What is the integral of (sin^6(x)) ?

The integral of (sin^6(x)) is -(cos(x)sin^5(x))/6+5/6 (-1/4 sin^3(x)cos(x)+3/8 (x-1/2 sin(2x)))+C

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integral of (sin^6(x))

\int (sin^{6} (x)) d x

- \frac{cos ( x ) sin ^{5} ( x )}{6} + \frac{5}{6} (- \frac{1}{4} sin^{3} (x) cos (x) + \frac{3}{8} (x - \frac{1}{2} sin (2 x))) + C

Solution steps

\int sin^{6} (x) d x

Apply Integral Reduction

= - \frac{cos ( x ) sin ^{5} ( x )}{6} + \frac{5}{6} \cdot \int sin^{4} (x) d x

\int sin^{4} (x) d x = - \frac{1}{4} sin^{3} (x) cos (x) + \frac{3}{8} (x - \frac{1}{2} sin (2 x))

= - \frac{cos ( x ) sin ^{5} ( x )}{6} + \frac{5}{6} (- \frac{1}{4} sin^{3} (x) cos (x) + \frac{3}{8} (x - \frac{1}{2} sin (2 x)))

Add a constant to the solution

= - \frac{cos ( x ) sin ^{5} ( x )}{6} + \frac{5}{6} (- \frac{1}{4} sin^{3} (x) cos (x) + \frac{3}{8} (x - \frac{1}{2} sin (2 x))) + C

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What is the integral of (sin^6(x)) ?
The integral of (sin^6(x)) is -(cos(x)sin^5(x))/6+5/6 (-1/4 sin^3(x)cos(x)+3/8 (x-1/2 sin(2x)))+C