can some make sense of this, where does the 5/4 I come from? i know you have to do some shifting because the last integral is "e^(-theta)cos2(theta)d(theta)" which is the same as the intial one.
Yes, your initial integral is called I. By using integration by parts the same integral came back in the righthandside of the equation. So you can now solve the equation for I, your initial integral. That's basically all.