Can I definite integral of $ \frac{e^{mx}}{x} dx $?

Question

I know about the exponential function $Ei(x)$, but that's just for indefinite integrals,right ?

Can I do a definite integral of $$\int_A^B \frac{e^{mx}}{x}\,dx$$ ?

Will it converge? Can you give me a function for the value?

Answer 1

You can still use ${\rm Ei}(x)$,

$$ {\rm Ei}(x) = -\int_{-x}^{+\infty}{\rm d}t~\frac{e^{-t}}{t} \tag{1} $$

Note that

\begin{eqnarray} \int_a^b{\rm d}x~\frac{e^{mx}}{x} &\stackrel{t = - mx}{=}& \int_{-ma}^{-mb}{\rm d}t~\frac{e^{-t}}{t} = \int_{-ma}^{+\infty}{\rm d}t~\frac{e^{-t}}{t} - \int_{-mb}^{+\infty}{\rm d}t~\frac{e^{-t}}{t}\\ &\stackrel{(1)}{=}& -{\rm Ei}(ma) + {\rm Ei}(mb) \end{eqnarray}