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