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

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

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

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

- \frac{1}{3} x cos^{3} (x) + \frac{2}{5} x cos^{5} (x) - \frac{1}{7} x cos^{7} (x) + \frac{8}{105} sin (x) + \frac{4}{315} sin^{3} (x) + \frac{1}{175} sin^{5} (x) - \frac{1}{49} sin^{7} (x) + C

Solution steps

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

Apply Integration By Parts

= x (- \frac{1}{3} cos^{3} (x) + \frac{2}{5} cos^{5} (x) - \frac{1}{7} cos^{7} (x)) - \int - \frac{1}{3} cos^{3} (x) + \frac{2}{5} cos^{5} (x) - \frac{1}{7} cos^{7} (x) d x

\int - \frac{1}{3} cos^{3} (x) + \frac{2}{5} cos^{5} (x) - \frac{1}{7} cos^{7} (x) d x = - \frac{8}{105} sin (x) + \frac{1}{7} sin^{3} (x) + \frac{1}{9} sin^{3} (x) - \frac{4}{15} sin^{3} (x) + \frac{2}{25} sin^{5} (x) - \frac{3}{35} sin^{5} (x) + \frac{1}{49} sin^{7} (x)

= x (- \frac{1}{3} cos^{3} (x) + \frac{2}{5} cos^{5} (x) - \frac{1}{7} cos^{7} (x)) - (- \frac{8}{105} sin (x) + \frac{1}{7} sin^{3} (x) + \frac{1}{9} sin^{3} (x) - \frac{4}{15} sin^{3} (x) + \frac{2}{25} sin^{5} (x) - \frac{3}{35} sin^{5} (x) + \frac{1}{49} sin^{7} (x))

Simplify

x (- \frac{1}{3} cos^{3} (x) + \frac{2}{5} cos^{5} (x) - \frac{1}{7} cos^{7} (x)) - (- \frac{8}{105} sin (x) + \frac{1}{7} sin^{3} (x) + \frac{1}{9} sin^{3} (x) - \frac{4}{15} sin^{3} (x) + \frac{2}{25} sin^{5} (x) - \frac{3}{35} sin^{5} (x) + \frac{1}{49} sin^{7} (x)) : - \frac{1}{3} x cos^{3} (x) + \frac{2}{5} x cos^{5} (x) - \frac{1}{7} x cos^{7} (x) + \frac{8}{105} sin (x) + \frac{4}{315} sin^{3} (x) + \frac{1}{175} sin^{5} (x) - \frac{1}{49} sin^{7} (x)

= - \frac{1}{3} x cos^{3} (x) + \frac{2}{5} x cos^{5} (x) - \frac{1}{7} x cos^{7} (x) + \frac{8}{105} sin (x) + \frac{4}{315} sin^{3} (x) + \frac{1}{175} sin^{5} (x) - \frac{1}{49} sin^{7} (x)

Add a constant to the solution

= - \frac{1}{3} x cos^{3} (x) + \frac{2}{5} x cos^{5} (x) - \frac{1}{7} x cos^{7} (x) + \frac{8}{105} sin (x) + \frac{4}{315} sin^{3} (x) + \frac{1}{175} sin^{5} (x) - \frac{1}{49} sin^{7} (x) + C

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