What is the integral of cos^2(ax) ?

The integral of cos^2(ax) is 1/2 (x+1/(2a)sin(2ax))+C

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integral of cos^2(ax)

\int cos^{2} (a x) d x

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

Solution steps

\int cos^{2} (a x) d x

Rewrite using trig identities

= \int \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 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 1 d x + \int cos (2 a x) d x)

\int 1 d x = x

\int cos (2 a x) d x = \frac{1}{2 a} sin (2 a x)

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

Add a constant to the solution

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

What is the integral of cos^2(ax) ?
The integral of cos^2(ax) is 1/2 (x+1/(2a)sin(2ax))+C