Question is to find the optimal value(s) (i.e., values of $x$ and corresponding $f$ values) of $\displaystyle f(x)=\int_{x}^{\infty}\frac{xs^{2}}{-1+e^{\alpha s}}~ds$ where $x>0$ and $\alpha>0$. If any one knows how to solve this analytically please explain. If analytical solution is not possible what sort of numerical method I can use?
I think Taylor series approximation cannot work as $s$ approaches infinity any polynomial approaches $\pm\infty$ while the integrand approaches zero. Am I correct here? Further, how safe it is to go inside the integral sign and differentiate when differentiate $f$ with respect to $x$ as the upper limit is infinite, if I am to do that?