Integration of $\int \frac{1}{x^{1/3}(x^{1/3}-1)}dx$

Question

Integrate the following function $$\int \frac{1}{x^{1/3}(x^{1/3}-1)}dx$$

Could someone give me slight hint to solve this question?

Answer 1

HINT:

Enforce the substitution $x=y^3$

SPOILER ALERT: Scroll over the highlighted area to reveal the solution

Enforcing the substitution $x=y^3$ yields$$\begin{align}\int \frac{1}{x^{1/3}(x^{1/3}-1)}\,dx&=3\int \frac{y}{y-1}\,dy\\\\&=3\left(x^{1/3}+\log\left(\left|x^{1/3}-1\right|\right)\right)+C\end{align}$$