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

The integral from-pi to pi of sin^2(nx) is pi-1/(2n)sin(2pin)

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

\int_{- π}^{π} sin^{2} (n x) d x

π - \frac{1}{2 n} sin (2 πn)

Solution steps

\int_{- π}^{π} sin^{2} (n x) d x

Rewrite using trig identities

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

\int_{- π}^{π} 1 d x = 2 π

\int_{- π}^{π} cos (2 n x) d x = \frac{sin ( 2 πn )}{n}

= \frac{1}{2} (2 π - \frac{sin ( 2 πn )}{n})

Simplify

\frac{1}{2} (2 π - \frac{sin ( 2 πn )}{n}) : π - \frac{1}{2 n} sin (2 πn)

= π - \frac{1}{2 n} sin (2 πn)

What is the integral from-pi to pi of sin^2(nx) ?
The integral from-pi to pi of sin^2(nx) is pi-1/(2n)sin(2pin)