The value of ∫ 0 12 π sin ⁡ t + - sin ⁡ t d t is equal to (where, . denotes the greatest integer function)

Step-by-step solution

We know that, x + - x = - 1 , x ∉ Z 0 , x ∈ Z So, ∫ 0 12 π sin ⁡ t + - sin ⁡ t d t = 6 ∫ 0 2 π sin ⁡ t + - sin ⁡ t d t = 6 ∫ 0 π 2 - 1 d t + ∫ π 2 π - 1 d t + ∫ π 3 π 2 - 1 d t + ∫ 3 π 2 2 π - 1 d t = 6 - 1 π 2 + π 2 + π 2 + π 2 = - 12 π