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

MCQ

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

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

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

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