First slide

Evaluation of definite integrals

Question

Let $I = \int_{- \infty}^{\infty} \frac{d x}{1 + x^{2}}$
Statement-1: $I = π$
Statement-2: The integrand is even and $\lim_{b \to \infty} F (b)$
$= π, F (b) = \int_{0}^{b} \frac{d x}{1 + x^{2}}$

Moderate

A

STATEMENT-1 is True, STATEMENT-2 is True; STATEMENT-2 is a correct explanation for STATEMENT-1
B

STATEMENT-1 is True, STATEMENT-2 is True; STATEMENT-2 is NOT a correct explanation for
STATEMENT-1
C

STATEMENT-1 is True, STATEMENT-2 is False
D

STATEMENT-1 is False, STATEMENT-2 is True

Solution

The integrand is even, so
$\int_{- \infty}^{\infty} \frac{d x}{1 + x^{2}} = 2 \int_{0}^{\infty} \frac{d x}{1 + x^{2}}$
$\int_{0}^{\infty} \frac{d x}{1 + x^{2}} = \lim_{b \to \infty} \int_{0}^{b} \frac{d x}{1 + x^{2}} = \lim_{b \to \infty} (\tan^{- 1} b - 0) = \frac{π}{2}$

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