Evaluating $\frac{2013^3-2\cdot 2013^2\cdot 2014+3\cdot 2013\cdot 2014^2-2014^3+1}{2013\cdot 2014}$

You can complete the cube of the difference.

$$ =\frac{2013^3-3\cdot2013^2\cdot2014+3\cdot2013\cdot2014^2-2014^3+2013^2\cdot2014+1}{2013\cdot2014} $$ $$ =\frac{(2013-2014)^3+2013^2\cdot2014+1}{2013\cdot2014}=\frac{2013^2\cdot2014}{2013\cdot2014}=2013 $$