What is the integral of 5cos^6(x)sin^2(x) ?

The integral of 5cos^6(x)sin^2(x) is

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integral of 5cos^6(x)sin^2(x)

\int 5 cos^{6} (x) sin^{2} (x) d x

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

Solution steps

\int 5 cos^{6} (x) sin^{2} (x) d x

Take the constant out:

\int a \cdot f (x) d x = a \cdot \int f (x) d x

= 5 \cdot \int cos^{6} (x) sin^{2} (x) d x

Rewrite using trig identities

= 5 \cdot \int cos^{6} (x) (1 - cos^{2} (x)) d x

Expand

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

= 5 \cdot \int cos^{6} (x) - cos^{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

= 5 (\int cos^{6} (x) d x - \int cos^{8} (x) d x)

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

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

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

Simplify

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

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

Add a constant to the solution

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

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What is the integral of 5cos^6(x)sin^2(x) ?
The integral of 5cos^6(x)sin^2(x) is