# How do you integrate (e^x/x)dx ?

Apr 12, 2017

This is sometimes called the exponential integral:

$\int {e}^{x} / x \mathrm{dx} = \text{Ei} \left(x\right) + C$

But the method I'd use (since I'm not familiar with the integral) is the Maclaurin series for ${e}^{x}$:

e^x=1+x+x^2/(2!)+x^3/(3!)+...=sum_(n=0)^oox^n/(n!)

Then:

e^x/x=1/x+1+x/(2!)+x^2/(3!)+...=1/x+sum_(n=0)^oox^n/((n+1)!)

So the antiderivative will be:

inte^x/xdx=int(1/x+1+x/(2!)+x^2/(3!)+...)dx=ln(absx)+x+x^2/(2*2!)+x^3/(3*3!)+...+C

inte^x/xdx=ln(absx)+sum_(n=1)^oox^n/(n*n!)+C