The integral to be evaluated is:

We define the function

and, by computing the fourth roots of -4,

we find that singularities

,

both lie inside the simple closed contour shown above, where . The other two singularities lie below the real axis. The theorem for finding rsidues at simple poles tells us

Note: Let two functions p and q be analytic at a point . If

, and

then is a simple pole of the quotient and

In our case:

since

we are now able to write

Furthermore, if z is a point in , then

where as

and this means that

Finally then