Question

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

Answer 1

Consider the given integral.

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

Put $t=\frac{x^{\frac{3}{2}}}{a^{\frac{3}{2}}}$

$dt=\frac{1}{a^{\frac{3}{2}}}\times \frac{3}{2}\sqrt{x}dx$

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

$I=\frac{2}{3}\int \frac{1}{\sqrt{1-t^2}}dt$

$I=\frac{2}{3}{\sin }^{-1}t+C$

On putting the value of $t$ in above expression, we get

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

Hence, this is the required value of the given integral.