If $$\int_0^{100\alpha} \frac{\sin^2 x}{e^{\left(\frac{x}{\pi} - \left[\frac{x}{\pi}\right]\right)}} dx = \frac{\alpha\pi^3}{1+4\pi^2}$$, $$\alpha \in R$$ where $$[x]$$ is the greatest integer less than or equal to $$x$$, then the value of $$\alpha$$ is:

A

$$200(1-e^{-1})$$

B

$$100(1-e)$$

C

$$50(e-1)$$

D

$$150(e^{-1}-1)$$