Power of sine!

Calculus Level 5

I(n)=0π2sin2nx dxX=r=1I(r)(2rr)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 104X\left\lfloor { 10 }^{ 4 }X \right\rfloor

Notation:

  • (nm)=n!n!(nm)! \displaystyle \binom{n}{m} = \frac{n!}{n! (n-m)!} denotes the binomial coefficient.
  • \lfloor \cdot \rfloor denotes the floor function.
This is part of my set Powers of the ordinary.
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