\[ \text { If } \begin{array}{l} \int_{0}^{1}\left(\sum_{r=1}^{2013} \frac{x}{x^{2}+r^{2}}\righ...

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\[
\text { If } \begin{array}{l}
\int_{0}^{1}\left(\sum_{r=1}^{2013} \frac{x}{x^{2}+r^{2}}\right)\left(\prod_{r=1}^{2013}\left(x^{2}+r^{2}\right)\right) d x \\
=\frac{1}{2}\left[\left(\prod_{r=1}^{2013}\left(1+r^{2}\right)\right)-k^{2}\right] \text { then } k=
\end{array}
\]
(a) 2013
(b) 2013 !
(c) \( 2013^{2} \)
(d) \( 2013^{2011} \)
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