I am not sure if I understood your question in the right way. So, you want to integrate the given function w.r.t.
Just substitute (and note that )
I need to calculate the expectation, which is the integral of xf(x). So this extra x cancels with the one in the denominator, and then all of that fraction can be taken out as its not dependent on x. To calculate the other part are you suggesting integration by substitution then?
The overall answer should be equal to sigma*e^mu though.