$\mathrm{Ei}(x)$, the exponential function, some question.

Question

I have a question involving with $\mathrm{Ei}(x)$, define as $\int_{-x}^{\infty}e^u \cdot u^{-1} \mathrm{d}u$.

My question is, when I have a expression say $\exp(x) \cdot \mathrm{Ei}(x)+1$. I want to optimize this function in terms of $x$. Now, the question is how do I suppose to with this $\mathrm{Ei}(x)$ function?

Answer 1

It may be helpful to note that $$\frac{d}{dt}\int^{\infty}_{t}f(u)du=-f(t)$$