Am I missing something with exponential Fourier series?

slashrulez
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Hi, I'm in a signal processing class, and I'm having some trouble with complex numbers. As an example, I've attached a pretty simple question about an exponential Fourier series.

I don't find these questions particularly hard, it just takes me ridiculously long to do them. I mean, after I take the integral, it literally took me 10 minutes to actually figure out how to get the general expression for different n values and whatnot. It just seems like it should be much easier, so I'm wondering if there's some kind of shortcut I'm missing.

My current method is pretty slow, and I just rewrite everything in trigonometric form and work from there. I can't think of any other way to do it, but it takes an awful long time. When I look at the work in that example, I don't really know what's going on. I don't know how they go to from the integral to that expression in one step, and I don't even know what the symbol attached to the npi/2 is.

Can anyone help me out? Thanks.
 
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