JEE Math Practice Question

If $$\int\limits_0^{100\pi } {{{{{\sin }^2}x} \over {{e^{\left( {{x \over \pi } - \left[ {{x \over \pi }} \right]} \right)}}}}dx = {{\alpha {\pi ^3}} \over {1 + 4{\pi ^2}}},\alpha \in R} $$ where [x] is the greatest integer less than or equal to x, then the value of $$\alpha$$ is :

1. A.200 (1 $$-$$ e$$-$$1)
2. B.100 (1 $$-$$ e)
3. C.50 (e $$-$$ 1)
4. D.150 (e$$-$$1 $$-$$ 1)