Find the following definite integral.

We are integrating the reciprocal of the function for the upper unit semicircle. If we look at both of those functions in the same graph

we can clnclude that the area of the gray shaded region is the same as the area of the green shaded region.

We see that our integral is asking for the area of an unbounded region. But while the integral going out to 1 is an unbounded region, the integral going out to any positive x which is less than 1 will be the area of a finite region and thus a finite number. The integral we are taking is the limit of these areas as x approaches 1. It turns out that this limit exists and is equal to pi/2.