Partial Fractions Integral

I have a feeling that I made my mistake in this question near the end, but I'll write it all out anyway:

equated co-efficients to get :

I can do the second and fourth integrals fine, so I'll just show work for the first and third.

First let

therefore

now I sub them in:

from the substitution I derived:

therefore

I'll stop here, because I think that by here I've made a mistake (this part of my answer does not match the correct answer). If anyone can spot it, it would be great.

Thanks,

Forgive typos, this one was quite tedious for me to write out.