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

The integral from-pi to pi of sin^2(ax) is pi-1/(2a)sin(2pia)

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

\int_{- π}^{π} sin^{2} (a x) d x

π - \frac{1}{2 a} sin (2 πa)

Solution steps

\int_{- π}^{π} sin^{2} (a x) d x

Rewrite using trig identities

= \int_{- π}^{π} 1 - cos^{2} (a 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_{- π}^{π} 1 d x - \int_{- π}^{π} cos^{2} (a x) d x

\int_{- π}^{π} 1 d x = 2 π

\int_{- π}^{π} cos^{2} (a x) d x = π + \frac{1}{2 a} sin (2 πa)

= 2 π - (π + \frac{1}{2 a} sin (2 πa))

Simplify

= π - \frac{1}{2 a} sin (2 πa)

What is the integral from-pi to pi of sin^2(ax) ?
The integral from-pi to pi of sin^2(ax) is pi-1/(2a)sin(2pia)