Question

Solve

${\int }^{}\frac{1}{x^{\frac{2}{3}}\sqrt{x^{\frac{2}{3}}-4}}dx$

Answer 1

$\int \frac{1}{x^{2/3}\sqrt{x^{\frac{2}{3}}-4}}dx$

Let $x^{1/3}=t$

On differentiating both sides, we have

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

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

$\Rightarrow \frac{1}{3}\int \frac{dt}{\sqrt{t^2-4}}=\int \frac{dt}{\sqrt{{\left(t\right)}^2-{\left(2\right)}^2}}$

$\Rightarrow \frac{1}{3}\ln (t+\sqrt{t^2-4})=\frac{1}{3}\ln (x^{\frac{1}{3}}+\sqrt{x^{\frac{2}{3}}-4})+C.$

Hence, solve.