Let t denote the greatest integer less than or equal to t . Then the value of the integral ∫ - 3 101 sin π x + e cos 2 π x d x is equal to

Step-by-step solution

Given, ∫ - 3 101 sin π x + e cos 2 π x d x Now let I 1 = ∫ - 3 101 sin π x d x and I 2 = ∫ - 3 101 e cos 2 π x d x Now checking periodicity of sin π x we get, 0 < x < 1 ⇒ sin π x = 0 1 < x < 2 ⇒ sin π x = - 1 sin π x → periodic with period 2 So, I 1 = 52 ∫ 0 2 sin π x d x = 52 ∫ 0 1 0 + ∫ 1 2 - 1 d x = - 52 Now checking periodicity of cos 2 π x we get, 0 < x < 1 4 ⇒ cos 2 π x =