# Integral of 1/((x)e^x) ?

## Homework Statement

Find the integral of

1/(xe^x)

## Homework Equations

None really... integration by parts maybe: integral of udv = vu - integral of vdu

## The Attempt at a Solution

I tried this by parts but didn't really get anywhere, it definitely doesn't simplify into anything useful. It's the solution to an ODE so maybe there's a type in my book, because I always get this as the final integral I have to compute.

hunt_mat
Homework Helper
Maybe that is the final answer to your question, it's an integral.

What was the ODE?

Char. Limit
Gold Member
If you want an analytic answer, you'll need to use the Exponential Integral function Ei(x), defined to be the integral from 0 to x of e^(t)/t dt.

EDIT: This can be written as $$Ei(x) = \int_0^x \frac{e^t}{t} dt$$

great, thank you

Dick
Homework Helper
If you want an analytic answer, you'll need to use the Exponential Integral function Ei(x), defined to be the integral from 0 to x of e^(t)/t dt.

EDIT: This can be written as $$Ei(x) = \int_0^x \frac{e^t}{t} dt$$

I think it would actually be an 'incomplete gamma function', since the e^t is in the denominator.

Char. Limit
Gold Member
I think it would actually be an 'incomplete gamma function', since the e^t is in the denominator.

Actually, if you write this as e^(-t)/t, and then substitute u=-t, du=-dt, you get this:

$$\int - \frac{e^u}{u} du$$

And the solution follows.

Dick
$$\int - \frac{e^u}{u} du$$