Answer
The integral $\int_{0}^{1} \dfrac{dx}{(xe^x+x^2)}$ diverges.
Work Step by Step
We are given the function
$f(x)=\int_{0}^{1} \dfrac{dx}{(xe^x+x^2)}$
Since, $x(e^x+x) \leq x(e+1)$
This yields:
$\dfrac{1}{x(e+1)} \leq \dfrac{1}{x(e^x+x)} $
But the integral $\int_{0}^{1} \dfrac{dx}{x(e+1)}$shows a p-type integral with $p=1$. Thus, the integral $\int_{0}^{1} \dfrac{dx}{x(e+1)}$ diverges.
Therefore, by the comparison test, the integral $\int_{0}^{1} \dfrac{dx}{(xe^x+x^2)}$ diverges as well.
