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$L=\int\limits_{0}^{1}x(x-1)^{2013}dx=\int\limits_{0}^{1}\left[ {(x-1)(x-1)^{2013}+(x-1)^{2013}} \right]dx$
$=\int\limits_{0}^{1}\left[ {(x-1)^{2014}+(x-1)^{2013}} \right]dx=\int\limits_{0}^{1}(x-1)^{2013}dx+\int\limits_{0}^{1}(x-1)^{2014}dx$
$=\left[ {\frac{(x-1)^{2014}}{2014}+\frac{(x-1)^{2015}}{2015}} \right]_{0}^{1}=-\frac{1}{2004}+\frac{1}{2005}$.
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