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BITSAT 2012

Exam: BITSAT
Year: 2012
Subject: Mathematics
Chapter: Indefinite Integration

BITSAT 2012 · Mathematics · Indefinite Integration

Evaluate: ∫ x 4- ^ 3 dx

Evaluate:

\int \sqrt{\frac{\mathrm{x}}{4-\mathrmx^{3}}} \mathrm{dx}

Options

A $\frac{2}{3} \sin ^{-1}\left(\frac{x^{3 / 2}}{2}\right)+c$
B $\frac{2}{3} \sin ^{-1}\left(x^{3 / 2}\right)+c$
C $2 \sin ^{-1}\left(\frac{x^{3 / 2}}{2}\right)+c$
D\frac{1}{3} \sin ^{-1} $\left(\frac{\mathrmx^{3 / 2}}{2}\right)+\mathrm{c}$

Correct answer

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

Step-by-step solution

I=\int \sqrt{\frac{x}{4-x^{3}}} d x=\int \frac{\sqrt{x} d x}{\sqrt{4-x^{3}}}

Here integral of

\sqrt{x}=\frac{2}{3} x^{3 / 2}

and

4-x^{3}=4-\left(x^{3 / 2}\right)^{2}

Put

x^{3 / 2}=t \Rightarrow \sqrt{x} d x=\frac{2}{3} d t

So

I=\frac{2}{3} \int \frac{d t}{\sqrt{4-t^{2}}}=\frac{2}{3} \sin ^{-1}\left(\frac{x^{3 / 2}}{2}\right)+c

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