ok. now, you do realize that the

in what you just worked out is a constant right? (we were integrating with respect to y). this constant represents

, a general complex number mentioned by Krizalid in post #36.

so that, your answer to this integral is in fact

now if we rationalize this, we get

. The imaginary part of this complex number is

, so that

as desired.

of course, the integral

was a LOT easier to compute than the by parts method. because

is simply a constant (complex constant, but a constant nonetheless)

one of the nice things about complex analysis is that a lot of the things in real calculus generalizes well with things in complex calculus, including how we integrate e raised to some constant times x, in general