# How to prove that the following function is monotonically increasing?

$$f(x) = \frac{\sqrt{x}\int_a^1 e^{-xs} s^{b+1}~ \mathrm{d}s}{\int_a^1 e^{-xs} s^{b} ~\mathrm{d}s}:\left]0,+\infty\right[\ \to \mathbb{R}$$

where $$0 < a < 1$$ and $$b > 0.$$

By applying elementary rules of differentiation I can only prove that the function is monotonically increasing in $$\left]\ 0,\frac{1}{1-a}\ \right]$$

But I need to prove that it is monotonically increasing in $$\left]\ 0,+\infty\ \right[$$

Your conjecture is incorrect. In particular, for $$a=b=1/10$$ we have $$f'(5)=-0.00561\ldots<0$$.