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

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

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

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

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

Solution steps

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

Apply Integration By Parts

: - cos^{2} (2 x) cos (x) - \int 2 sin (4 x) cos (x) d x

= - cos^{2} (2 x) cos (x) - \int 2 sin (4 x) cos (x) d x

\int 2 sin (4 x) cos (x) d x = - \frac{1}{5} cos (5 x) - \frac{1}{3} cos (3 x)

= - cos^{2} (2 x) cos (x) - (- \frac{1}{5} cos (5 x) - \frac{1}{3} cos (3 x))

Simplify

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

Add a constant to the solution

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

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