We can't just integrate cos^2(x) as it is, so we want to change it into another form, which we can easily do using trig identities.

Recall the double angle formula: cos(2x) = cos^2(x) - sin^2(x). We also know the trig identity sin^2(x) + cos^2(x) = 1, so combining these we get the equation cos(2x) = 2cos^2(x) -1.

Now we can rearrange this to give: cos^2(x) = (1+cos(2x))/2.
So we have an equation which gives cos^2(x) in a nicer form which we can easily integrate using the reverse chain rule.

This eventually gives us an answer of x/2 + sin(2x)/4 +c
:)