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.

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