The following integral has a singularity at $t = 0$ as in this situation the exponential term becomes $1$ and it no longer dominates the $\frac{1}{t}$ term.
$$f(x) = \int_0^1\frac{1}{t}e^{-t}dt$$
So is it possible to analytically or numerically integrate this function?
The singularity isn't integrable, so you will have $+\infty$, as you can see by observing $\int_0^1 e^{-t}/t dt \geq \int_0^{1} \frac{1}{et} dt = +\infty$.
The question of numerically integrating $\int_0^1 t^a e^{-t}$ for $-1<a<0$ is more interesting.