Question

*$\int \sqrt{\left(\frac{x}{a^3-x^3}\right)\cdot }dx=\frac{6}{k}{\sin }^{-1}\frac{x^{3/2}}{a^{3/2}}.$ Find the value of $k$.

Answer 1

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