# Iterated Integration with a fraction

• Millenniumf
In summary, the conversation is about a user struggling with iterated integration, specifically with the equation ∫ylnx/x from e^y to y for dx, y>0. The user mentions being out of practice with alternate methods for integration and not being able to solve the equation using u-substitution. Another user suggests a substitution of u=ln(x), which helps with the fraction in the equation. The original user realizes their mistake and thanks the other user for their help.

#### Millenniumf

I need help with Iterated Integration. I'm trying to integrate the following equation:

∫ylnx/x from e^y to y for dx, y>0

(sorry about the messy equation; first time user)

I know how to integrate ln, but the fraction has me thrown for a loop. I tried U-substitution, but the equation kept getting messier. To be clear, I'm out of practice with some of the alternate methods for integration, so I may not be considering something (I planned on practicing them over the summer, but I got a job and my time got eaten away).

Did you try the rather obvious substitution: u= ln(x)?

Yes; it's the fraction I'm having a problem with, though. I am not sure why, though. I've done them before. This semester though I haven't run into them often, and the ones I've seen were done in class as example problems (the teacher picks from his homework assignments).

I do have integration tables in the back of the book. I'll try those and see if one of them matches.

Hello Millenniumf,

Welcome to PF.

The substitution suggested by HallsofIvy takes care of the "fraction" very nicely.

$\displaystyle \int\frac{\ln(x)}{x}dx$

Let u=ln(x), then du = ?

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Huh, it's obvious now! u=ln(x), du=(1/x), then I just input e^y and y and complete the problem.

I think my brain must have been fried from pulling an all-nighter getting my assignment completed; I've done more complex u-sub problems before with no complications.

I think I better start earlier in the day with future assignments.

Thank you for the help!