When you say the standard contour what do you mean? You need to integrate on a vertical line that is to the right of the real part of all of the poles of the integrand.
One such vertical line is
The imaginary axis.
This gives the integral
This is now an integral with z real (an integral on the real axis) edit: sorry this was a bad choice of variables
Now we can apply the residue theorem on the semicircle above the real axis and the line segment on the real axis.
So by the residue theorem we get
Now the reason we wanted to write the integral this way
is so we can apply Jordan's lemma
We can identify
and note that
So by Jordan's Lemma Jordan's Lemma -- from Wolfram MathWorld
and we are done.