Integration by Parts

you are correct up to the point where you write out uv - int(vdu)

to evaluate int( x^2 / ( x^2 - 1 ) ), use partial fractions to split the bottom into linear terms, do a u substitution to move the 1's into the numerator instead of the denominator, and then evaluate the resulting simple integrals. but DO NOT split or multiply integrals the way you did in lines 5 & 6.

very tedious, very torturous, i wrote down about half the work and don't have a math program / word 2010 on this computer. diligence is all you need.

 Spoiler

final answer should be:
x ln(x^2 -1) - 2x + ln(x-1) - ln(x+1) + c

whoa, the ninjas are out early today...

lol, yeah i didnt saw what kalovale did, i thought that he did algebraic integration at x^2/(x-1)(x+1) dx after setting the by parts method