Is There a Solution to This Tricky Integral? 1/(xe^x)

[S.N.]

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]

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

What was the ODE?

 

[Char. Limit]

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 $$Ei(x) = \int_0^x \frac{e^t}{t} dt$$

 

[S.N.]

great, thank you

 

[Dick]

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]

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]

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.

True. The incomplete gamma of degree 0 is basically the same as the Ei. You can represent it either way. Sorry.