What is the integral from-pi to pi of cos^2(mx) ?

The integral from-pi to pi of cos^2(mx) is pi+1/(2m)sin(2pim)

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integral from-pi to pi of cos^2(mx)

\int_{- π}^{π} cos^{2} (m x) d x

π + \frac{1}{2 m} sin (2 πm)

Solution steps

\int_{- π}^{π} cos^{2} (m x) d x

Rewrite using trig identities

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

\int_{- π}^{π} 1 d x = 2 π

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

= \frac{1}{2} (2 π + \frac{sin ( 2 πm )}{m})

Simplify

\frac{1}{2} (2 π + \frac{sin ( 2 πm )}{m}) : π + \frac{1}{2 m} sin (2 πm)

= π + \frac{1}{2 m} sin (2 πm)

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What is the integral from-pi to pi of cos^2(mx) ?
The integral from-pi to pi of cos^2(mx) is pi+1/(2m)sin(2pim)