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If \(\displaystyle I=\int _{ 0 }^{ 1 }{ { \left( \ln { x } \right) }^{ 100 } \, dx } \) , then find the trailing number of zeros in \(I\).

\[ \large \sum_{r=0}^{502} (-1)^r \sin\left( \dfrac{(2r+1)\pi}{2014} \right) = \, ? \]

If \(\displaystyle \large \int _{ 0 }^{ 1 }{ { x }^{ 2015 }{ e }^{ -x }dx } =2015!-k\sum _{ r=0 }^{ 2015 }{ \left( ^{ 2015 }{ { C }_{ r } }\times \left( r! \right) \right) } \), find ...

\[ \large \int_0^\infty (-\{ x \} )^{\lfloor x \rfloor} \, dx = \ ?\]

Notation:

\( \{ x\} \) denote the fractional part of \(x\).

\( \lfloor x\rfloor \) denote the floor function.

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