Question

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

Answer 1

Solution :-

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

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

Let $x^{3/2}=t$

$\Rightarrow \frac{3}{2}{x}^{\frac{3}{2}-1}dx=dt$

$\Rightarrow \frac{3}{2}{x}^{1/2}dx=dt$

$\Rightarrow x^{1/2}dx=\frac{2}{3}dt$

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

$=\frac{2}{3}si{n}^{-1}\frac{t}{a^{3/2}}+c$

$=\frac{2}{3}si{n}^{-1}\frac{x^{3/2}}{a^{3/2}}+c$