Can $\int_0^1 \frac{1}{x} e^{-x} dx$ be integrated?

Question

I have an integral with a singularity at $x = 0$.

$$\int_0^1 \frac{1}{x} e^{-x} dx$$

It's not a removable singularity so is it possible to perform the integration? For example could some complex analysis technique or some numeric method be used to integrate this? Or is it simply not integrable?

Answer 1

This is not integrable because $\int_0^1 \frac{1}{x}e^{-x} dx > \frac{1}{e}\int_0^1 \frac{1}{x} dx = \infty$