Question

Find $\int \frac{\sqrt{x}}{\sqrt{a^3-x^3}}dx$

Answer 1

$I=\int \frac{\sqrt{x}}{\sqrt{a^3-x^3}}dx$

Let $t=\frac{x}{a}$

$\Rightarrow dt=\frac{dx}{a}$

$I=\int a\sqrt{\frac{at}{a^3-a^3{t}^3}}dt$

$I=\int \sqrt{\frac{t}{1-t^3}}dt$

Let $1-t^3=s^2$

$\Rightarrow -3t^{2}dt=2sds$

$\Rightarrow I=\int \frac{-2}{3\sqrt{1-s^2}}ds$

$\Rightarrow I=\frac{-2}{3}{\sin }^{-1}s+c$

$I=\frac{-2\times {\sin }^{-1}\left(\sqrt{1-\frac{x^3}{a^3}}\right)}{3}$