Power of sine!

Calculus Level 5

$I(n)= \int _{ 0 }^{ \frac { \pi }{ 2 } }{ \sin ^{ 2n }{ x } } \ dx \qquad \qquad X=\sum _{ r=1 }^{ \infty }{ \frac { I(r) }{ { \left( \begin{matrix} 2r \\r \end{matrix} \right) } } }$

Given the above, find $\left\lfloor { 10 }^{ 4 }X \right\rfloor$

Notation:

• $\displaystyle \binom{n}{m} = \frac{n!}{n! (n-m)!}$ denotes the binomial coefficient.
• $\lfloor \cdot \rfloor$ denotes the floor function.
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