Part of your problem is that for the integral, you can't use half the interval and multiply by two like you could for the case. You can only do that, in general, when your complete integrand is even. But the presence of that particular exponential function in the integrand precludes that. So you have to keep the full interval for the limits in the integration, I'm afraid.
I don't know of any way to integrate other than by parts twice. I'm not sure I would recommend breaking the integrals up into the trig functions, because then you're going to have to do integration by parts twice on two integrals. There's no need, when working with the exponential Fourier series, to look at sin and cos individually. It's an easier integration with just the exponential in there.
So I would carry these changes through, and then we'll see what happens at the end. Sound good?