First slide

Evaluation of definite integrals

Question

Let $I = \int_{0}^{π} x^{2} \cos x d x and J = \int_{0}^{π} x \sin x d x$
Statement-1: $I = - 2 π$
Statement-2: $I = 2 J$

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

Integrating by parts
$\int_{0}^{π} x^{2} \cos x = {x^{2} \sin x|}_{0}^{π} - \int_{0}^{π} 2 x \sin x d x = - 2 J$
$J = \int_{0}^{π} x \sin x d x = \int_{0}^{π} π \sin x d x - \int_{0}^{π} x \sin x d x$
$\Rightarrow 2 J = π (- \cos x) ‖_{0}^{π} = 2 π \Rightarrow J = π$

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