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JEE Main · 2023 · 24 Jan 2023 · Shift 1 · Definite Integrals

\int_{\frac{3 2}{4}}^{\frac{3 3}{4}} \frac{48}{9 - 4 x ^{2}} d x

is equal to

(A) $\frac{π}{3}$
(B) $\frac{π}{2}$
(C) $\frac{π}{6}$
(D) $2 π$ ✓ Correct

Answer: (D)

Solution

Writing

9 - 4 x^{2} = 9 - (2 x)^{2}

with

a = 3

and using

\int \frac{d x}{a ^{2} - u ^{2}} = sin^{- 1} \frac{u}{a}

with

u = 2 x

, the antiderivative is

48 \cdot \frac{1}{2} sin^{- 1} \frac{2 x}{3} = 24 sin^{- 1} \frac{2 x}{3}

. At

x = \frac{3 3}{4}

,

\frac{2 x}{3} = \frac{3}{2}

and

sin^{- 1} \frac{3}{2} = \frac{π}{3}

; at

x = \frac{3 2}{4}

,

\frac{2 x}{3} = \frac{1}{2}

and

sin^{- 1} \frac{1}{2} = \frac{π}{4}

. The integral is

24 (\frac{π}{3} - \frac{π}{4}) = 24 \cdot \frac{π}{12} = 2 π

.

JEE Main · 2023 · 24 Jan 2023 · Shift 1 · Single correct option

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