Can you help me to get the integration of $\frac{1}{x} \exp(-x)$?

Question

I need the solution of integral like $$\int^\infty_a \frac{1}{x} e^{-x} \,dx.$$

Thank you

Answer 1

This is the definition of the exponential integral $$ \int \frac{e^{-x}}{x} dx=Ei(-x). $$ Thus we can write $$ \int_a^\infty \frac{e^{-x}}{x} dx=-Ei(-a),\quad \Re(a)>0. $$ There is nothing to prove, it is just definition of the function. See here http://en.wikipedia.org/wiki/Exponential_integral, where they write $$ -\int_{-x}^\infty \frac{e^{-x}}{x} dx=Ei(x). $$ Let me know if this helps.

Answer 2

This is a so-called exponential integral and cannot be put in closed form in general, see also here for example.

Answer 3

What you are asking for is exactly the Exponential Integral: $-Ei(x)$

You can look at it on MathWorld. It doesn't looks like a integral that could be exactly solved.