Question

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

Answer 1

$I=\int \frac{\sqrt{x}}{\sqrt{a^2-x^3}}dx$

$=\int \frac{\sqrt{x}}{\sqrt{(a{\frac{3}{2}}^2-(x{\frac{3}{2}}^2)}}dx$

Now let $x^{\frac{3}{2}}=t$

$\frac{3}{2}\sqrt{x}dx=dt$

Substituting $\sqrt{x}dx=\frac{2}{3}dt$ in $I$

$I=\frac{2}{3}\int \frac{dt}{\sqrt{(a^{\frac{3}{2}}{)}^2-t^2}}$

$I=\frac{2}{3}{\sin }^{-1}\frac{-1}{a^{\frac{3}{2}}}$

$=\frac{2}{3}{\sin }^{-1}\frac{x^{\frac{3}{2}}}{a^{\frac{3}{2}}}$