Question

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

Answer 1

Let $I=\int \frac{\sqrt{x}}{\sqrt{a^3-x^3}}dx=\int \frac{\sqrt{x}}{\sqrt{(a^{3/2}{)}^2-(x^{3/2}{)}^2}}Put{x}^{3/2}=t\Rightarrow \frac{3}{2}{x}^{1/2}dx=dt\therefore I=\frac{2}{3}\int \frac{dt}{\sqrt{(a^{3/2}{)}^2-t^2}}=\frac{2}{3}si{n}^{-1}\frac{t}{a^{3/2}}+C=\frac{2}{3}si{n}^{-1}\frac{x^{3/2}}{a^{3/2}}+C=\frac{2}{3}si{n}^{-1}\sqrt{\frac{x^3}{a^3}}+C$