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

The integral of (sin^2(2x)+sin(2x)) is 1/2 (x-1/4 sin(4x))-1/2 cos(2x)+C

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

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

\frac{1}{2} (x - \frac{1}{4} sin (4 x)) - \frac{1}{2} cos (2 x) + C

Solution steps

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

Apply the Sum Rule:

\int f (x) \pm g (x) d x = \int f (x) d x \pm \int g (x) d x

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

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

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

= \frac{1}{2} (x - \frac{1}{4} sin (4 x)) - \frac{1}{2} cos (2 x)

Add a constant to the solution

= \frac{1}{2} (x - \frac{1}{4} sin (4 x)) - \frac{1}{2} cos (2 x) + C

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What is the integral of (sin^2(2x)+sin(2x)) ?
The integral of (sin^2(2x)+sin(2x)) is 1/2 (x-1/4 sin(4x))-1/2 cos(2x)+C