If the integral ∫ 0 10 sin 2 π x e x - x d x = α e - 1 + β e - 1 2 + γ , where α , β , γ are integers and x denotes the greatest integer less than or equal to x , then the value of α + β + γ is equal to:

Step-by-step solution

Let I = ∫ 0 10 sin 2 π x e x - x d x = ∫ 0 10 sin 2 π x e x d x , where x - x = x is the fractional part function. We know that the x is a periodic function with period 1 and also that sin 2 π x is a periodic function with period 2 π 2 π = 1 . Thus, the function f x = sin 2 π x e x is periodic with period 1 . If f x is a periodic function with period T then ∫ 0 n T f x d x = n ∫ 0 T f x d x ,   n ∈ N Therefore I = 10 ∫ 0 1 sin 2 π x e x d x