What is the integral of sin^2(x)sin(3x) ?

The integral of sin^2(x)sin(3x) is -1/3 sin^2(x)cos(3x)+1/6 (-1/5 cos(5x)+cos(x))+C

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integral of sin^2(x)sin(3x)

\int sin^{2} (x) sin (3 x) d x

- \frac{1}{3} sin^{2} (x) cos (3 x) + \frac{1}{6} (- \frac{1}{5} cos (5 x) + cos (x)) + C

Solution steps

\int sin^{2} (x) sin (3 x) d x

Apply Integration By Parts

: - \frac{1}{3} sin^{2} (x) cos (3 x) - \int - \frac{1}{3} sin (2 x) cos (3 x) d x

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

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

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

Simplify

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

Add a constant to the solution

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

\frac{\partial}{\partial x} (x + 4)

\frac{1}{2 s ^{3} + 6 s ^{2} + 4 s}

\int x (x) d x

y = (3 x + 4)^{5} + 15 x (3 x + 4)^{4}

\int - e^{4 x} d x

What is the integral of sin^2(x)sin(3x) ?
The integral of sin^2(x)sin(3x) is -1/3 sin^2(x)cos(3x)+1/6 (-1/5 cos(5x)+cos(x))+C