How to integrate $\int \cos^2(3x)dx$ $$\int \cos^2(3x)dx$$
The answer according to my instructor is:
$${1 + \cos(6x) \over 2} + C$$
But my book says that:
$$\int \cos^2(ax)dx = {x \over 2} + {\sin(2ax) \over 4a} + C$$
I'm not really sure which one is correct. 

What your instructor told you is $\cos^2(ax) = \frac{1+\cos(2ax)}{2}$ (half angle formula), then you can integrate it easily.
