IGNOU PHE 14 2026 - Using the method of residues, show that:

Question

Using the method of residues, show that:

$$\int_{0}^{\infty} \frac{dx}{x^2 + 1} = \frac{\pi}{2}$

29 Jan 2026

Answer :

Word Count : 174

Consider the integral [ I = \int_0^{\infty} \frac{dx}{x^2 + 1}. ] We extend it to the entire real axis: [ \int_{-\infty}^{\infty} \frac{dx}{x^2 + 1} = 2 \int_0^{\infty} \frac{dx}{x^2 + 1} = 2 I. ] Now consider the complex function [ f(z) = \frac{1}{z^2 + 1}

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