First slide

Evaluation of definite integrals

Question

If $\int_{0}^{\infty} e^{- x^{2}} d x = \frac{\sqrt{π}}{2}$ then
Statement-1: $\int_{0}^{\infty} \frac{e^{- x}}{\sqrt{x}} d x = \sqrt{π}$
Statement-2: $lim_{x \to \infty} e^{- x^{2}} = 0$

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

Put $x = t^{2}$
$\int_{0}^{\infty} \frac{e^{- x}}{\sqrt{x}} d x = 2 \int_{0}^{\infty} \frac{e^{- t^{2}}}{t} t d t = 2 \int_{0}^{\infty} e^{- t^{2}} d t$

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