If \({\int }_{0}^{100\pi }\frac{{\sin }^{2}x}{{e}^{\left(\frac{x}{\pi }-\left[\frac{x}{\pi }\right]\right)}}dx=\frac{a{\…

If \({\int }_{0}^{100\pi }\frac{{\sin }^{2}x}{{e}^{\left(\frac{x}{\pi }-\left[\frac{x}{\pi }\right]\right)}}dx=\frac{a{\pi }^{3}}{1+4{\pi }^{2}},a\in R\), where \([x]\) is the greatest integer less than or equal to \(x\), then the value of \(\alpha\) is:

[JEE Main 2021, 22 Jul (Shift 2)]