Integrals: Evaluating ∫ x^{m}* (ln(x))^{2} using Integration by Parts

[Justabeginner]

Homework Statement

Evaluate the integral: ∫ x^{m}* (ln(x))^{2} It's said as ln squared x. Sorry if I miswrote it.

Homework Equations

∫udv= uv - ∫vdu

The Attempt at a Solution

u= (ln(x)^2)
v= x^{m+1}/(m+1)
du= 2lnx/(x)
dv= x^{m} * dx

- ∫2lnx * (x^{m+1})/(x*(m+1)) + [(ln(x)^{2}) (x^{m+1})/(m+1)}
-(x^{m+1})/(m+1) * ∫ 2lnx/(x) + (ln(x)^{2})

I am stuck here. I feel like I'm not doing this right, and I'm sure I'm not. Can I get some guidance as to if I'm even doing this right? Thank you so much.

 

[vela]

Up to here, you're okay:
$$\int x^m (\ln x)^2\,dx = (\ln x)^2\frac{x^{m+1}}{m+1} - \frac{2}{m+1} \int x^m\ln x\,dx.$$ Your next step is wrong. You can't pull anything that depends on $x$ out of the integral.

Did you notice you've ended up with essentially the same integral as before except the exponent of the log has gone done by one? This suggests you should try integrating by parts one more time.