​ If ∫ sin^2x/e^(x/π-[x/π]) dx x ∈ [0,100π] = απ^3/1+4π^2 α ∈ R where [x] is the greatest integer less than or equal to x, then the value of α is:

Question

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

∫ sin²x/e^{(x/π-[x/π])}dx x ∈ [0, 100π] = απ³/1+4π²

(1) 200 (1 – e^–1 )

(2) 100 (1 – e)

(3) 50 (e – 1)

(4) 150 (e^–1 –1)

Answer 1

Answer is: (1) 200 (1 – e^-1)