Let Ι = ∫ 0 24 π sin ⁡ x d x , then the value of 2 I is equal to (where, . denotes the fractional part function)

Step-by-step solution

Applying a + b - x property, Ι = ∫ 0 24 π s i n 24 π - x d x = ∫ 0 24 π - sin ⁡ x d x Adding the two integrals, we get, 2 Ι = ∫ 0 24 π sin ⁡ x + - sin ⁡ x d x Now, α + - α = 0 ; α ∈ Ι 1 ; α ∉ Ι As, sin ⁡ x takes integral values only at discrete points, Hence, 2 Ι = ∫ 1 0 24 π d x = x 0 24 π = 24 π - 0 = 24 π