Perturbation applied to electron wave function in a solid

I've been working on what happens when the positive ions in the lattice of a solid are taken into account in what is know as the Nearly Free Electron approach. Perturbation theory is the idea that you slightly wobble the steady state energy level and see what happens. Here the perturbation is given a value V and is modelled by a Fourier series because it is periodic, rising and falling from one ion to the next. Applying first order perturbation theory you get this for the final change to the energy

 I had to swot up on my maths to work out why the integrals circled come out to zero.

 For the next bit, I read that you only get non-zero solutions for the values of k given.

 I looked up the integral for cos-squared and I'm hoping that my cancelling is allowed.Then end result is 1/2V.