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

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

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

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

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

Solution steps

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

Rewrite using trig identities

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

Apply Integration By Parts

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

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

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

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

Simplify

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

Add a constant to the solution

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

sin (tan (9 x))

\int \frac{1}{1 ^{2} + 2 x ^{2}} d x

\int \frac{x ^{3} - 5 x ^{2} + 3 x + 14}{x - 3} d x

f (x) = e^{- 3 x}, (0, 1)

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