Question

Let $\int \frac{x^{1 / 2}}{\sqrt{1-x^{3/2}}} d x=\frac{2}{3} \operatorname{gof}(x)+C$, then

• A. $f ( x )=\sqrt{ x }$
• B. $f ( x )= x ^{3 / 2}$ and $g ( x )=\sin ^{-1} x$
• C. $f ( x )= x ^{2 / 3}$
• D. None of these

Answer 1

Answer: (B)

Put $x ^{3 / 2}= t \Rightarrow \frac{3}{2} x ^{1 / 2} dx = dt$
$\therefore $ integral is
$\int \frac{\frac{2}{3} dt }{\sqrt{1- t ^{2}}}=\frac{2}{3} \sin ^{-1} t + C$
$=\frac{2}{3} \sin ^{-1}\left( x ^{3 / 2}\right)+ C$