Homework Help: Integrating, probably by parts

1. Oct 26, 2006

mbrmbrg

I have the expression $$\int{x(\ln{x})^3dx}$$
I thought I had a quick way to integrate by parts but it turned out that I had accidentally evaluated $$\int{x\ln{x}dx}$$ instead.
Revisiting $$\int{x(\ln{x})^3dx}$$, I wanted to start by making a strange substitution, wherein u=ln(x), du=1/x dx, and x=e^u. This meant that when I rewrote the integral, instead of multiplying dx by a constant to get it to be du, I multiplied it by x (which in this case was e^u). Is that allowed? Because I got a very different, much uglier answer than the book's.

I'd appreciate any comments, whether on my weird "method" or on a more standard approach to evaluating $$\int{x(\ln{x})^3dx}$$

2. Oct 26, 2006

wurth_skidder_23

Try integration by parts with u = (ln(x))^3 and dv = x dx

Last edited: Oct 26, 2006
3. Oct 26, 2006

Max Eilerson

Your substitution method should work fine. Your should be integrating $$\int{Exp[2u] u^3du}$$. If you do it by integration by parts, you will need to do it 3 times.

4. Oct 26, 2006

mbrmbrg

thanks, that got me the book's answer!

5. Oct 27, 2006

mbrmbrg

And yes, the other way does work also. Nifty!

Share this great discussion with others via Reddit, Google+, Twitter, or Facebook