What is the integral from 0 to 1 of cos^2(ax) ?

The integral from 0 to 1 of cos^2(ax) is 1/2 (1+(sin(2a))/(2a))

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integral from 0 to 1 of cos^2(ax)

\int_{0}^{1} cos^{2} (a x) d x

\frac{1}{2} (1 + \frac{sin ( 2 a )}{2 a})

Solution steps

\int_{0}^{1} cos^{2} (a x) d x

Rewrite using trig identities

= \int_{0}^{1} \frac{1 + cos ( 2 a 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_{0}^{1} 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

= \frac{1}{2} (\int_{0}^{1} 1 d x + \int_{0}^{1} cos (2 a x) d x)

\int_{0}^{1} 1 d x = 1

\int_{0}^{1} cos (2 a x) d x = \frac{sin ( 2 a )}{2 a}

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

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What is the integral from 0 to 1 of cos^2(ax) ?
The integral from 0 to 1 of cos^2(ax) is 1/2 (1+(sin(2a))/(2a))