Improper Integrals with the Residue Theorem: Example 2

We give our next application of the Residue Theorem. We compute a real improper integral interpreted in the Cauchy principal value sense. We extend the integrand, which is the function 1/(1+x^n) for a natural number n, to the complex function, f(z) = 1/(1+z^n). We integrate this function over the the sector of the circle of radius R with angle 2pi/n. The two lines in this contour are multiples of the integral we wish to find. The portion on the arc tends to zero as R tends to infinity. An application of the residue theorem yields the result. #mikethemathematician, #profdabkowski, #mikedabkowski, #complexanalysis, #improper