I am looking for an examination of the convergence/divergence of the integral alone without calculation of the integral: $$\int_0^1 \frac {2x^{1/3}+3\sin^2x} {e^{\tan x^2}-1}dx$$
By and large it seems to me that there is divergence of the integral due to $x=0$. However, how to prove that the integral does not convergence in a clear way?
Near $x=0$, the denominator is $\sim\tan(x^2)\sim x^2$, and the numerator is $\sim 2x^{1/3}$. So the integrand is $\sim 2x^{-5/3}$ and hence the integral diverges there, as $\int_0^\varepsilon x^p\,\mathrm{d}x$ converges iff $p>-1$.