Let [t] denote the largest integer less than or equal to t . If ₀^3 ( [x^2 ]+ [ x^2 2 ] ) d x= a + b 2 - 3 - 5 + c 6 - 7 , where a , b , c Z , then a + b + c is equal to_______

Step-by-step solution

aligned & ₀^3 [x^2 ] d x+ ₀^3 [ x^2 2 ] d x & = ₀^1 0 d x+ ₁¹² 1 d x+ _ 2 ^ 3 2 d x aligned aligned & + _ 3 ^2 3 dx + ₂^ 5 4 dx + _ 5 ^ 6 5 dx & + _ 6 ^ 7 6 dx + _ 7 ^ 8 7 dx + _ 8 ^3 8 dx & + ₀^ 2 0 dx + _ 2 ^2 1 dx & + ₂^ 6 2 dx + _ 6 ^ 8 3 dx + _ 8 ^3 4 dx =31-6 2 - 3 - 5 & -2 6 - 7 & a =31 ~b =-6 c =-2 & a + b + c =31-6-2=23 aligned