**31.** Use a rectangular contour to evaluate the integrals:

**(a) **

The upper side of the rectangle should be at
(for real
Then on the upper side:

Then around the whole rectangle:

Along the end at
,
with

provided that

Along the end at

provided
that
Thus we have:

Now
the integrand has a pole where

or

which
is inside the contour. The residue there may be found from method 4:

and
so the integral is:

and
thus

The
result is real, as expected.