WBJEE · Maths · Indefinite Integration
Let \(\int \frac{x^{1 / 2}}{\sqrt{1-x^3}} d x=\frac{2}{3} g(f(x))+c\); then
- A \(f(x)=\sqrt{x}, g(x)=x^{3 / 2}\)
- B \(f(x)=x^{3 / 2}, g(x)=\sin ^{-1} x\)
- C \(f(x)=\sqrt{x}, g(x)=\sin ^{-1} x\)
- D \(\quad f(x)=\sin ^{-1} x, g(x)=x^{3 / 2}\)
Answer & Solution
Correct Answer
(B) \(f(x)=x^{3 / 2}, g(x)=\sin ^{-1} x\)
Step-by-step Solution
Detailed explanation
\(\int \frac{x^{1 / 2} d x}{\sqrt{1-\left(x^{3 / 2}\right)^2}}=2 / 3 \int \frac{d t}{\sqrt{1-t^2}}\) \(x^{3 / 2}=t\) \(3 / 2 x^{1 / 2} d x=d t\) \(=2 / \sin ^{-1}(t)+c\) \(=2 / 3 \sin ^{-1}\left(x^{3 / 2}\right)+c\)
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