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

The integral of cos^6(θ)sin^2(θ) is

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integral of cos^6(θ)sin^2(θ)

\int cos^{6} (θ) sin^{2} (θ) d θ

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

Solution steps

\int cos^{6} (θ) sin^{2} (θ) d θ

Rewrite using trig identities

= \int cos^{6} (θ) (1 - cos^{2} (θ)) d θ

Expand

cos^{6} (θ) (1 - cos^{2} (θ)) : cos^{6} (θ) - cos^{8} (θ)

= \int cos^{6} (θ) - cos^{8} (θ) d θ

Apply the Sum Rule:

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

= \int cos^{6} (θ) d θ - \int cos^{8} (θ) d θ

\int cos^{6} (θ) d θ = \frac{sin ( θ ) cos ^{5} ( θ )}{6} + \frac{5}{6} (\frac{1}{4} cos^{3} (θ) sin (θ) + \frac{3}{8} (θ + \frac{1}{2} sin (2 θ)))

\int cos^{8} (θ) d θ = \frac{sin ( θ ) cos ^{7} ( θ )}{8} + \frac{7}{8} (\frac{sin ( θ ) cos ^{5} ( θ )}{6} + \frac{5}{6} (\frac{1}{4} cos^{3} (θ) sin (θ) + \frac{3}{8} (θ + \frac{1}{2} sin (2 θ))))

= \frac{sin ( θ ) cos ^{5} ( θ )}{6} + \frac{5}{6} (\frac{1}{4} cos^{3} (θ) sin (θ) + \frac{3}{8} (θ + \frac{1}{2} sin (2 θ))) - (\frac{sin ( θ ) cos ^{7} ( θ )}{8} + \frac{7}{8} (\frac{sin ( θ ) cos ^{5} ( θ )}{6} + \frac{5}{6} (\frac{1}{4} cos^{3} (θ) sin (θ) + \frac{3}{8} (θ + \frac{1}{2} sin (2 θ)))))

Simplify

\frac{sin ( θ ) cos ^{5} ( θ )}{6} + \frac{5}{6} (\frac{1}{4} cos^{3} (θ) sin (θ) + \frac{3}{8} (θ + \frac{1}{2} sin (2 θ))) - (\frac{sin ( θ ) cos ^{7} ( θ )}{8} + \frac{7}{8} (\frac{sin ( θ ) cos ^{5} ( θ )}{6} + \frac{5}{6} (\frac{1}{4} cos^{3} (θ) sin (θ) + \frac{3}{8} (θ + \frac{1}{2} sin (2 θ))))) : \frac{5}{192} cos^{3} (θ) sin (θ) + \frac{1}{6} cos^{5} (θ) sin (θ) + \frac{5}{32} (2 θ + sin (2 θ)) - \frac{1}{8} cos^{7} (θ) sin (θ) - \frac{7}{48} cos^{5} (θ) sin (θ) - \frac{35}{256} (2 θ + sin (2 θ))

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

Add a constant to the solution

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