# Integrals-Extra Variables

1. May 25, 2013

### Justabeginner

1. The problem statement, all variables and given/known data
Evaluate the integral: ∫ $x^{m}* (ln(x))^{2}$ It's said as ln squared x. Sorry if I miswrote it.

2. Relevant equations
∫udv= uv - ∫vdu

3. 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.

2. May 25, 2013

### vela

Staff Emeritus
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.