Answer
$(3) \ln |\dfrac{\sqrt[3] {x}}{1+\sqrt[3] {x}}|+c$
Work Step by Step
Consider the integral $\int \dfrac{dx}{x(1+\sqrt[3] {x})}$
Let us take the help of the substitution method.
$a=\sqrt[3] {x} \implies da=\dfrac{dx}{3x^{2/3}}; dx=3a^2 da$
The given integral can be re-written as: $\int \dfrac{dx}{x(1+\sqrt[3] {x})}=(3) \int \dfrac{da}{a(1+a)}$
This implies that $(3) \int \dfrac{da}{a(1+a)}=(3) \int(\dfrac{1}{a}- \dfrac{1}{(1+a)})da$
We use the formula: $\int x^n dx=\int\dfrac{x^{n+1}}{n+1} dx$
or, $(3)\ln | \dfrac{a}{1+a}|+C=(3) \ln |\dfrac{\sqrt[3] {x}}{1+\sqrt[3] {x}}|+c$
