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

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

Popular >

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

\int_{- π}^{π} sin^{2} (2 x) d x

π

Solution steps

\int_{- π}^{π} sin^{2} (2 x) d x

Rewrite using trig identities

= \int_{- π}^{π} \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

= \frac{1}{2} \cdot \int_{- π}^{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

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

\int_{- π}^{π} 1 d x = 2 π

\int_{- π}^{π} cos (4 x) d x = 0

= \frac{1}{2} (2 π - 0)

Simplify

\frac{1}{2} (2 π - 0) : π

= π

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

\int_{0}^{4} 2 x + 1 d x

\int_{0}^{\frac{π}{2}} 2 cos^{2} (θ) d θ

\int_{0}^{2} 1 - x^{2} d x

\int_{0}^{\frac{5}{8}} \frac{1}{64 x ^{2} + 25} d x

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