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

The integral of sin^2(2x)sin(5x) is -1/5 sin^2(2x)cos(5x)+1/45 (-cos(9x)+9cos(x))+C

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

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

- \frac{1}{5} sin^{2} (2 x) cos (5 x) + \frac{1}{45} (- cos (9 x) + 9 cos (x)) + C

Solution steps

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

Apply Integration By Parts

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

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

\int - \frac{2}{5} sin (4 x) cos (5 x) d x = - \frac{1}{45} (- cos (9 x) + 9 cos (x))

= - \frac{1}{5} sin^{2} (2 x) cos (5 x) - (- \frac{1}{45} (- cos (9 x) + 9 cos (x)))

Simplify

= - \frac{1}{5} sin^{2} (2 x) cos (5 x) + \frac{1}{45} (- cos (9 x) + 9 cos (x))

Add a constant to the solution

= - \frac{1}{5} sin^{2} (2 x) cos (5 x) + \frac{1}{45} (- cos (9 x) + 9 cos (x)) + C

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What is the integral of sin^2(2x)sin(5x) ?
The integral of sin^2(2x)sin(5x) is -1/5 sin^2(2x)cos(5x)+1/45 (-cos(9x)+9cos(x))+C