∫ 0 2 2 x 2 - 3 x + x - 1 2 d x , where t is the greatest integer function, is equal to

Step-by-step solution

∫ 0 2 2 x 2 - 3 x + x - 1 2 d x = ∫ 0 2 2 x 2 - 3 x d x + ∫ 0 2 x - 1 2 d x Let I 1 = ∫ 0 2 2 x 2 - 3 x d x = ∫ 0 3 2 3 x - 2 x 2 d x + ∫ 3 2 2 2 x 2 - 3 x d x = 3 x 2 2 - 2 x 3 3 0 3 2 + 2 x 3 3 - 3 x 2 2 3 2 2 = 27 8 - 9 4 + 8 3 - 6 - 9 4 + 27 8 = 9 4 - 10 3 = 19 12 Let I 2 = ∫ 0 2 x - 1 2 d x Assume x - 1 2 = t i.e. I 2 = ∫ - 1 2 3 2 t d t = ∫ - 1 2 0 - 1 d t + ∫ 0 1 0 · d t + ∫ 1 3 2 1 · d t = - t - 1 2 0 + t 1 3 2 = - 1 2 + 3 2 - 1 = 0