What is the integral from 0 to 2pi of (sin(2x))^2 ?

The integral from 0 to 2pi of (sin(2x))^2 is pi

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integral from 0 to 2pi of (sin(2x))^2

\int_{0}^{2 π} (sin (2 x))^{2} d x

π

Decimal Notation

3.14159 \dots

Solution steps

\int_{0}^{2 π} (sin (2 x))^{2} d x

Rewrite using trig identities

= \int_{0}^{2 π} 4 cos^{2} (x) sin^{2} (x) d x

Take the constant out:

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

= 4 \cdot \int_{0}^{2 π} cos^{2} (x) sin^{2} (x) d x

Rewrite using trig identities

= 4 \cdot \int_{0}^{2 π} \frac{1 - cos ( 4 x )}{8} d x

Take the constant out:

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

= 4 \cdot \frac{1}{8} \cdot \int_{0}^{2 π} 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

= 4 \cdot \frac{1}{8} (\int_{0}^{2 π} 1 d x - \int_{0}^{2 π} cos (4 x) d x)

\int_{0}^{2 π} 1 d x = 2 π

\int_{0}^{2 π} cos (4 x) d x = 0

= 4 \cdot \frac{1}{8} (2 π - 0)

Simplify

4 \cdot \frac{1}{8} (2 π - 0) : π

= π

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What is the integral from 0 to 2pi of (sin(2x))^2 ?
The integral from 0 to 2pi of (sin(2x))^2 is pi