​ Evaluate : ∫ √x/1 - x^3 dx; x ∊ (0,1). ​

Question

Evaluate :

∫ \(\sqrt\frac{x}{1\ -\ x^3}\) dx; x ∊ (0,1).

Answer 1

Let \(x^\frac{3}{2}\) = t

⇒ \(\frac{3}{2}\)\(x^\frac{3}{2}\)dx

∫\(\sqrt\frac{x}{1\ -\ x^3}\)dx

= \(\frac{2}{3}\)∫\(\frac{dt}{\sqrt{1\ -\ t^2}}\)

= \(\frac{2}{3}\) sin^-1(t) + c

= \(\frac{2}{3}\)sin^-1\((x^\frac{3}{2})\) + c, where 'c' is an arbitrary constant of integration.