If the value of the integral ∫ 0 5 x + [ x ] e x - [ x ] d x = α e - 1 + β , where α , β ∈ R , 5 α + 6 β = 0 , and x denotes the greatest integer less than or equal to x ; then the value of ( α + β ) 2 is equal to :

Step-by-step solution

∫ 0 5 x + [ x ] e x - [ x ] d x = α e - 1 + β ⇒ ∫ 0 1 x e x d x + ∫ 1 2 x + 1 e x - 1 d x + ∫ 2 3 x + 2 e x - 2 d x + ∫ 3 4 x + 3 e x - 3 d x + ∫ 4 5 x + 4 e x - 4 d x Let x - 1 = p ,   x - 2 = q ,   x - 3 = r ,   x - 4 = w ⇒ ∫ 0 1 x e x d x + ∫ 0 1 p + 2 e p d p + ∫ 0 1 q + 4 e q d q + ∫ 0 1 r + 6 e r d r + ∫ 0 1 w + 8 e w d w ⇒ ∫ 0 1 x e x d x + ∫ 0 1 x + 2 e x d x + ∫ 0 1 x + 4 e x d x +