1. Let so that
Hence we have which may or may not be a standard integral. I'll outline the integration below:
We know from our trig identities that so let's sub that into the integral:
We know that so then the reverse must be true: (the constant will be added later)
Thus where C is the constant of integration.
Back substitute to get your integral in terms of x
2. I'd go with a double angle identity:
which can be rearranged to give
In your example and hence .
The integral of cos(ax) is for constant, non-zero a.