Integrating, probably by parts

[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 $$\int{x(\ln{x})^3dx}$$

 

[wurth_skidder_23]

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

 

[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.

 

[mbrmbrg]

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

thanks, that got me the book's answer!

 

[mbrmbrg]

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