\[ \begin{array}{l} \int_{-2011}^{2011}\left\{\left[x^{2011}+x^{2009}+x^{2007}+x^{2005}+\ldots+x....

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\[
\begin{array}{l}
\int_{-2011}^{2011}\left\{\left[x^{2011}+x^{2009}+x^{2007}+x^{2005}+\ldots+x\right]\right\} d x \\
+\int_{-201}^{201}\left[\left\{x^{2010}+x^{2008}+x^{2006}+\ldots+1\right\}\right] d x \text { is equal }
\end{array}
\]
\( \mathrm{P} \)
\( +\int_{-201}^{2011}\left[\left\{x^{2010}+x^{2008}+x^{2006}+\ldots+1\right\}\right] d x \) is equal
to (where [.] represents the greatest integer function and \( \{ \).\( \} represents the fractional part \) function)
(1) zero
(2) \( 2 \times(2011) \) !
(3) \( 2^{2011} \)
(4) \( 2 \times(2010) \) !


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