What is the integral of sin^6(x)cos^2(x) ?

The integral of sin^6(x)cos^2(x) is

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integral of sin^6(x)cos^2(x)

\int sin^{6} (x) cos^{2} (x) d x

- \frac{5}{192} sin^{3} (x) cos (x) - \frac{1}{6} sin^{5} (x) cos (x) + \frac{5}{32} (2 x - sin (2 x)) + \frac{1}{8} sin^{7} (x) cos (x) + \frac{7}{48} sin^{5} (x) cos (x) - \frac{35}{256} (2 x - sin (2 x)) + C

Solution steps

\int sin^{6} (x) cos^{2} (x) d x

Rewrite using trig identities

= \int sin^{6} (x) (1 - sin^{2} (x)) d x

Expand

sin^{6} (x) (1 - sin^{2} (x)) : sin^{6} (x) - sin^{8} (x)

= \int sin^{6} (x) - sin^{8} (x) d x

Apply the Sum Rule:

\int f (x) \pm g (x) d x = \int f (x) d x \pm \int g (x) d x

= \int sin^{6} (x) d x - \int sin^{8} (x) d 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)))

\int sin^{8} (x) d x = - \frac{cos ( x ) sin ^{7} ( x )}{8} + \frac{7}{8} (- \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))))

= - \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))) - (- \frac{cos ( x ) sin ^{7} ( x )}{8} + \frac{7}{8} (- \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)))))

Simplify

- \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))) - (- \frac{cos ( x ) sin ^{7} ( x )}{8} + \frac{7}{8} (- \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))))) : - \frac{5}{192} sin^{3} (x) cos (x) - \frac{1}{6} sin^{5} (x) cos (x) + \frac{5}{32} (2 x - sin (2 x)) + \frac{1}{8} sin^{7} (x) cos (x) + \frac{7}{48} sin^{5} (x) cos (x) - \frac{35}{256} (2 x - sin (2 x))

= - \frac{5}{192} sin^{3} (x) cos (x) - \frac{1}{6} sin^{5} (x) cos (x) + \frac{5}{32} (2 x - sin (2 x)) + \frac{1}{8} sin^{7} (x) cos (x) + \frac{7}{48} sin^{5} (x) cos (x) - \frac{35}{256} (2 x - sin (2 x))

Add a constant to the solution

= - \frac{5}{192} sin^{3} (x) cos (x) - \frac{1}{6} sin^{5} (x) cos (x) + \frac{5}{32} (2 x - sin (2 x)) + \frac{1}{8} sin^{7} (x) cos (x) + \frac{7}{48} sin^{5} (x) cos (x) - \frac{35}{256} (2 x - sin (2 x)) + C

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