Question

Solve:

$\int \frac{\sqrt{1-x^3}}{x^2\sqrt{x}}dx$ ?

Answer 1

$\int \frac{\sqrt{1-x^3}}{x^2\sqrt{x}}dx=(\frac{1}{x\sqrt{x}}-\frac{5}{3x^{\frac{3}{2}}})\sqrt{1-x^3}-\int \sqrt{\frac{x}{1-x^3}}dx\left[applyingintegrationbyparts\right]=(\frac{1}{x\sqrt{x}}-\frac{5}{3x^{\frac{3}{2}}})\sqrt{1-x^3}-\frac{2}{3}{\sin }^{-1}\left(x^{\frac{3}{2}}\right)+C$