First slide

Introduction to definite integration

Question

Let $I = \int_{1}^{2} \frac{10 x^{2}}{{(x^{3} + 1)}^{2}} d x and J = \int_{2}^{9} \frac{d x}{x^{2}}$
Statement-1: $I = \frac{35}{27}$
Statement-2: $3 I = 10 J$

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

$Set x^{3} + 1 = u$
$\Rightarrow \int_{1}^{2} \frac{10 x^{2}}{{(x^{3} + 1)}^{2}} d x = \frac{10}{3} \int_{2}^{9} \frac{d u}{u^{2}} = \frac{10}{3} {[- \frac{1}{u}]}_{2}^{9} = \frac{35}{27} .$

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