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

The integral of cos(2x)cos^2(5x) is 1/4 sin(2x)+1/20 cos(2x)sin(10x)+1/160 sin(8x)-1/240 sin(12x)+C

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

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

\frac{1}{4} sin (2 x) + \frac{1}{20} cos (2 x) sin (10 x) + \frac{1}{160} sin (8 x) - \frac{1}{240} sin (12 x) + C

Solution steps

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

Apply Integration By Parts

: \frac{1}{2} cos (2 x) (x + \frac{1}{10} sin (10 x)) - \int - sin (2 x) (x + \frac{1}{10} sin (10 x)) d x

= \frac{1}{2} cos (2 x) (x + \frac{1}{10} sin (10 x)) - \int - sin (2 x) (x + \frac{1}{10} sin (10 x)) d x

\int - sin (2 x) (x + \frac{1}{10} sin (10 x)) d x = \frac{1}{2} x cos (2 x) - \frac{1}{4} sin (2 x) - \frac{1}{160} sin (8 x) + \frac{1}{240} sin (12 x)

= \frac{1}{2} cos (2 x) (x + \frac{1}{10} sin (10 x)) - (\frac{1}{2} x cos (2 x) - \frac{1}{4} sin (2 x) - \frac{1}{160} sin (8 x) + \frac{1}{240} sin (12 x))

Simplify

\frac{1}{2} cos (2 x) (x + \frac{1}{10} sin (10 x)) - (\frac{1}{2} x cos (2 x) - \frac{1}{4} sin (2 x) - \frac{1}{160} sin (8 x) + \frac{1}{240} sin (12 x)) : \frac{1}{4} sin (2 x) + \frac{1}{20} cos (2 x) sin (10 x) + \frac{1}{160} sin (8 x) - \frac{1}{240} sin (12 x)

= \frac{1}{4} sin (2 x) + \frac{1}{20} cos (2 x) sin (10 x) + \frac{1}{160} sin (8 x) - \frac{1}{240} sin (12 x)

Add a constant to the solution

= \frac{1}{4} sin (2 x) + \frac{1}{20} cos (2 x) sin (10 x) + \frac{1}{160} sin (8 x) - \frac{1}{240} sin (12 x) + C

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What is the integral of cos(2x)cos^2(5x) ?
The integral of cos(2x)cos^2(5x) is 1/4 sin(2x)+1/20 cos(2x)sin(10x)+1/160 sin(8x)-1/240 sin(12x)+C