The integral $\int \frac{dx}{\sqrt{9-4x^{2}}}$ is equal to

(B) $\frac{1}{2}\sin^{-1}\left(\frac{2x}{3}\right)+c$

(A) $\frac{1}{6}\sin^{-1}\left(\frac{2x}{3}\right)+c$

(C) $\sin^{-1}\left(\frac{2x}{3}\right)+c$

(D) $\frac{3}{2}\sin^{-1}\left(\frac{2x}{3}\right)+c$

CBSE Class 12 · 2024 · Indefinite IntegralsMultiple Choice

The integral

\int \frac{d x}{9 - 4 x ^{2}}

is equal to

Answer: (B)

Write

9 - 4 x^{2} = 3^{2} - (2 x)^{2}

. With

u = 2 x

\int \frac{d u}{9 - u ^{2}} = sin^{- 1} \frac{u}{3}

, and the factor

\frac{1}{2}

from

d u = 2 d x

gives

\frac{1}{2} sin^{- 1} (\frac{2 x}{3}) + C .

Hence option (B).

CBSE Class 12 Board · 2024 · Single correct option