x/(1*2)+x/(2*3)+.+x/(2011*2012)=2011
x[1/(1*2)+1/(2*3)+.+1/(2011*2012)]=2011
x(1-1/2+1/2-1/3+.+1/2011-1/2012)=2011
x(1-1/2012)=2011
2011x/2012=2011
x/2012=1
x=2012