Question

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

Answer 1

This is sometimes called the exponential integral:

`inte^x/xdx="Ei"(x)+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`