Q59 / 155 · WBJEE 2022 · Mathematics

Let $\int \frac{x^{1 / 2}}{\sqrt{1-x^3}} d x=\frac{2}{3} g(f(x))+c$; then

  • A
    $f(x)=\sqrt{x}, g(x)=x^{3 / 2}$
  • Correct answer
    B
    $f(x)=x^{3 / 2}, g(x)=\sin ^{-1} x$
  • C
    $f(x)=\sqrt{x}, g(x)=\sin ^{-1} x$
  • D
    $\quad f(x)=\sin ^{-1} x, g(x)=x^{3 / 2}$

Solution

$\int \frac{x^{1 / 2} d x}{\sqrt{1-\left(x^{3 / 2}\right)^2}}=2 / 3 \int \frac{d t}{\sqrt{1-t^2}}$ $x^{3 / 2}=t$ $3 / 2 x^{1 / 2} d x=d t$ $=2 / \sin ^{-1}(t)+c$ $=2 / 3 \sin ^{-1}\left(x^{3 / 2}\right)+c$

Hence, the correct answer is option: 'B'.

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