Intergration by Parts (IbP) problem

[Ravenatic20]

$$\int_{1} ^{2e} x^2(ln x)^{2} dx$$

I need to solve this using IbP. I made the following:

$$u = (ln x)^2$$

$$du = (\frac{2 ln x}{x}) dx$$



$$dv = x^2 dx$$

$$v = \frac{x^3}{3}$$

So I get:
$$(ln x)^2 (\frac{x^3}{3}) \|_{1} ^{2e}$$$$- \int_{1} ^{2e} (\frac{x^3}{3}) 2 ln x dx$$
(not sure how to make this look right)

Is this right?
Where do I go from here? Thanks

 

[awvvu]

Integrate by parts again.

 

[Ravenatic20]

So what I have is right so far? I just IbP on the RHS?

 

[awvvu]

Yup, if you've integrated ln(x) before, it should be obvious that you can integrate it again.