If the definite integral I = ∫ 0 π sin ⁡ x d x = ∑ ε = 0 n a ε sin ⁡ ε . (where, ⋅ is the greatest integer function), then the value of a n + n π is equal to

Step-by-step solution

The given integral is I = ∫ 0 1 sin ⁡ 0 d x + ∫ 1 2 sin ⁡ 1 d x + ∫ 2 3 sin ⁡ 2 d x + ∫ 3 π sin ⁡ 3 d x ⇒ I = 2 - 1 sin ⁡ 1 + 3 - 2 sin ⁡ 2 + π - 3 sin ⁡ 3 ∴ a n = π - 3 and n = 3 ⇒ a n + n = π or a n + n π = 1