The value of ∫ - π / 2 π / 2 d x x + sin ⁡ x + 4 , where t denotes the greatest integer less than or equal to t , is

Step-by-step solution

We have ∫ - π 2 π 2 d x [ x ] + [ s i n x ] + 4 = ∫ - π 2 0 d x [ x ] + 3 + ∫ 0 π 2 d x [ x ] + 4 = ∫ - π 2 - 1 d x 1 + ∫ - 1 0 d x 2 + ∫ 0 1 d x 4 + ∫ 1 π 2 d x 5 = [ x ] - π 2 - 1 + x 2 - 1 0 + x 4 0 1 + x 5 1 π 2 = - 1 + π 2 + 0 + 1 2 + 1 4 + π 10 - 1 5 = - 20 + 10 π + 10 + 5 + 2 π - 4 20 = 12 π - 9 20 = 3 20 4 π - 3