First slide

Methods of integration

Question

$\int \sqrt{\frac{x}{1 - x^{3}}} dx$ is equal to

Easy

A

$\frac{2}{3} \sin^{- 1} (x^{23}) + C$
B

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

$\frac{3}{2} \sin^{- 1} (x^{23}) + C$
D

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

Solution

$I = \int \sqrt{\frac{x}{1 - x^{3}}} dx = \int \frac{\sqrt{x}}{\sqrt{1 - x^{3}}} dx = \int \frac{\sqrt{x}}{\sqrt{1 - {(x^{3 / 2})}^{2}}} dx$
put $x^{3 / 2} = t \Rightarrow \sqrt{x} dx = \frac{2}{3} dt$
so, $I = \frac{2}{3} \int \frac{dt}{\sqrt{1 - t^{2}}} = \frac{2}{3} \sin^{- 1} (x^{3 / 2}) + C$

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