How do I integrate cos^2(x)?

The key to solving any integral of this form is to use the cosine rule:

cos(2x) = cos²(x) - sin²(x) = 2cos²(x) - 1 = 1 - 2sin²(x)

All of these forms are really helpful when solving problems such as this, and it's great if you can remmeber them, though if you get stuck in an exam, they can all be derived from the addition formulae that are probably on your fomula sheet!

So, using the above idenities, we know that:

2cos²(x) - 1 = cos(2x)

2cos²(x) = cos(2x) + 1

cos²(x) = (cos(2x) + 1)/2

So instead, we perform the integral of (cos(2x) + 1)/2, which we already know how to do.

=> (sin(2x))/4 + x/2