Originally Posted by

**astuart** Hello,

I have a pretty simple integration by parts question I'm supposed to solve, but for some reason I can't seem to get the right answer...Can someone point out where I've gone wrong?

$\displaystyle \int x\, ln x \, dx$

So, $\displaystyle u = ln x \, $

$\displaystyle dv= x \, dx \, $

$\displaystyle du= \frac {1}{x} \,dx $

$\displaystyle v = \frac {x^2}{2}$

Our tutor instructed us that if there is a natural log in the question, that that should be set as the variable to be deriviated, as we haven't been taught a technique for finding the integral of ln x.

Anyway, I worked the question out as follows...

$\displaystyle = uv - \int v \, du$

$\displaystyle = ln\,x (\frac {x^2}{2}) - \int \frac {x^2}{2} \, \frac {1}{x} \, dx $

$\displaystyle = \frac {x^2 ln \,x}{2} - \frac {1}{2} \int x^2 \, \frac {1}{x} \,dx$

$\displaystyle = \frac {x^2 ln \,x}{2} - \frac {1}{2} \cdot \frac {x^3}{3} \, ln\,x\,+\,C$