Evaluate: x a^3-x^3 d x - Vidyadip

MCQ

Evaluate: $\int \frac{\sqrt{x}}{\sqrt{a^3-x^3}} d x$

A
$\frac{3}{2} \sin ^{-1}\left(\frac{x}{a}\right)^{3 / 2}+C$
✓
$\frac{2}{3} \sin ^{-1}\left(\frac{x}{a}\right)^{3 / 2}+C$
C
$\frac{2}{3} \cos ^{-1}\left(\frac{x}{a}\right)^{3 / 2}+C$
D
$\frac{3}{2} \cos ^{-1}\left(\frac{x}{a}\right)^{3 / 2}+C$

✓

Answer

Correct option: B.

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

Let $I=\int \frac{\sqrt{x}}{\sqrt{a^3-x^3}} d x$
Put $x^{3 / 2}=t \Rightarrow \frac{3}{2} x^{1 / 2} d x=d t$
$\therefore I=\frac{2}{3} \int \frac{d t}{\sqrt{a^3-t^2}}$
$=\frac{2}{3} \int \frac{d t}{\sqrt{\left(a^{3 / 2}\right)^2-t^2}}$
$=\frac{2}{3}\left[\sin ^{-1}\left(\frac{t}{a^{3 / 2}}\right)\right]+C$
$=\frac{2}{3}\left[\sin ^{-1}\left(\frac{x^{3 / 2}}{a^{3 / 2}}\right)\right]+C$
$=\frac{2}{3} \sin ^{-1}\left(\frac{x}{a}\right)^{3 / 2}+C$

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