Answer
The integral $\int_{0}^{\infty} \dfrac{dx}{(x+x^2)^{1/3}}$ diverges.
Work Step by Step
We are given the function
$f(x)=\int_{0}^{\infty} \dfrac{dx}{(x+x^2)^{1/3}}$
Since, $(x+x^2)^{1/3} \geq x^2$
This yields:
$\dfrac{dx}{(x+x^2)^{1/3}} \geq \dfrac{1}{(2x^2)^{1/3}} $
Consider the integral $\int_{1}^{\infty} \dfrac{dx}{(2x^2)^{1/3}}= \dfrac{1}{2^{1/3}} [3x^{1/3}]_1^{\infty}\\=\infty$
Thus, the integral $\int_{1}^{\infty} \dfrac{dx}{(2x^2)^{1/3}}$ diverges. Therefore, by the comparison test, the integral $\int_{0}^{\infty} \dfrac{dx}{(x+x^2)^{1/3}}$ diverges as well.
