What is the integral of-2pisin^2(2x) ?

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

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integral of-2pisin^2(2x)

\int - 2 π sin^{2} (2 x) d x

- π (x - \frac{1}{4} sin (4 x)) + C

Solution steps

\int - 2 π sin^{2} (2 x) d x

Take the constant out:

\int a \cdot f (x) d x = a \cdot \int f (x) d x

= - 2 π \cdot \int sin^{2} (2 x) d x

Rewrite using trig identities

= - 2 π \cdot \int \frac{1}{2} (1 - cos (4 x)) d x

Take the constant out:

\int a \cdot f (x) d x = a \cdot \int f (x) d x

= - 2 π \frac{1}{2} \cdot \int 1 - cos (4 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

= - 2 π \frac{1}{2} (\int 1 d x - \int cos (4 x) d x)

\int 1 d x = x

\int cos (4 x) d x = \frac{1}{4} sin (4 x)

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

Simplify

- 2 π \frac{1}{2} (x - \frac{1}{4} sin (4 x)) : - π (x - \frac{1}{4} sin (4 x))

= - π (x - \frac{1}{4} sin (4 x))

Add a constant to the solution

= - π (x - \frac{1}{4} sin (4 x)) + C

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