Simply this exponential integral

Question

I am trying to simplify of this:

$$\int_{0}^{\infty} \frac{1-e^{-x}}{x}e^{-\lambda x}\,dx.$$

Maybe I should separate these equation into two exponential integral function? But it will ended up with infinite minus infinite? please give me some help or advices, thanks!

Answer 1

Write ${1-e^{-x}\over x}$ as $\int_0^1 e^{-tx}\,dt$, reverse the order of integration in the resulting double integral, then integrate.