Using the method I of contour integration, evaluate integrate (x ^ 2)/((x ^ 2 + 1) ^ 2) dx from - ∞ to ∞
To evaluate the integral using contour integration, we will consider the contour integral of the function f(z) = (z ^ 2)/((z ^ 2 + 1) ^ 2) around a semicircle in the upper half plane.
Let's define the contour integral C as the integral over the contour, which consists of the x-axis from -R to R and the semicircular arc in the upper half plane from R to -R as R goes to infinity and a small semi-circle of radius ε around z = i.
The residue of the function f(z) at the point z = i is given by:
Res(f, i) = lim(z->i) ((z-i) * (z+i) ^ 2) / ((z ^ 2 + 1) ^ 2)
= lim(z->i) ((z-i) * (z-i) ^ 2) / ((z-i) ^ 2 * (z + i) ^ 2)
= -i/4
According to the residue theorem, the contour integral around the closed curve is equal to 2 * π * i times the sum of residues inside the contour. Since we only have one residue inside the contour (Res(f, i) = -i/4), the integral is:
C = 2 * π * i * (-i/4) = π/2
Since the integral along the small semi-circle around z = i approaches 0 (as ε approaches 0), and the integral along the large semi-circle in the upper half plane approaches 0 (as R approaches infinity), we are left with the following integral along the x-axis:
∫ [-(R to R)] (x ^ 2)/((x ^ 2 + 1) ^ 2) dx = π/2
This is the integral that we originally wanted to evaluate. Therefore, the value of the integral ∫ (x ^ 2)/((x ^ 2 + 1) ^ 2) dx from - ∞ to ∞ is π/2.