JEE Main · 2019 · 11 Jan 2019 · Shift 1 · Indefinite Integrals

\int \frac{1 - x ^{2}}{x ^{4}} d x = A (x) (1 - x^{2})^{m} + C

, for a suitable chosen integer

m

and a function

A (x)

, where

C

is a constant of integration, then

(A (x))^{m}

equals:

Answer: (A)

Solution

Rewrite the integrand as

\frac{1}{x ^{3}} \frac{1}{x ^{2}} - 1

and substitute

t = \frac{1}{x ^{2}} - 1

, so

d t = - \frac{2}{x ^{3}} d x

. The integral becomes

- \frac{1}{2} \int t d t = - \frac{1}{3} t^{3/2} = - \frac{1}{3} (\frac{1}{x ^{2}} - 1)^{3/2} = - \frac{1}{3 x ^{3}} (1 - x^{2})^{3}

. Matching

A (x) (1 - x^{2})^{m}

gives

m = 3

and

A (x) = - \frac{1}{3 x ^{3}}

, so

(A (x))^{m} = (- \frac{1}{3 x ^{3}})^{3} = - \frac{1}{27 x ^{9}}

JEE Main · 2019 · 11 Jan 2019 · Shift 1 · Single correct option

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