How to solve e^x lnx dx

Homework Statement

∫e^x lnx dx
I dont really know how to solve it.

The Attempt at a Solution

This is what i have:
∫e^x lnxdx = lnx e^x - ∫e^x/x dx

And my prof says its wrong, that i can go further with it with some method they discussed about ( i missed it :(

maajdl
Gold Member
But you are right.
What you did is not wrong, but it might not be the answer to the exact question that was asked.
And ∫e^x/x dx is related to the function called Ei for "exponential integral". http://en.wikipedia.org/wiki/Exponential_integral.

There is maybe a misunderstanding.
Could you explain the statement of the problem more completely?
Is it only about calculating this integral?
Or is it about using some method to develop this integral in some way (like a series)?
Was the problem written black on white on paper?

I was just given this integral, nothing else.
Maybe they calculeted this in another way, or used something that im not aware in the class, but i was only given this one example.
Prof says that what i've got is okay, but its not about what they did in the class, what possibly could it be?

maajdl
Gold Member
So you missed some classes! Bad boy!!
Your Prof tries to punish you, isn't it?
Or eventually, give us a reference to the textbook that your Prof uses.
There are many different ways to evaluate this integral, guessing without information is total loss of time.
You have the right to know what the question exactly is.
Don't accept such a stupid game.

He uses Mathematical Techniques by Jordan & Smith ( Oxford)
He dont want to tell me exactly what method i should use, my classmates dont remember.
SO

D H
Staff Emeritus
I haven't the foggiest idea what your professor is saying is wrong, but ##\int e^x \ln x\, dx## is indeed ##e^x \ln x - \int \frac{e^x}{x} dx##. That integral on the right hand side? That's essentially the exponential integral (plus an arbitrary constant).So another way of expressing this is that ##\int e^x \ln x\, dx = e^x \ln x - \operatorname{Ei}(x) + \text{constant}##.

HallsofIvy
Homework Helper
The reason why "Ei(x)" is given a special name like that is that $\int\frac{e^x}{x}dx$ is not any "elementary" function.

He said i need to get at least two of that integrals that i've got. So maybe he mean that i need to do it again by parts, right ?

Ray Vickson