Let [x] denote the greatest integer ≤ x . Consider the function f ( x ) = max x 2 , 1 + [ x ] . Then the value of the integral ∫ 0 2 f ( x ) d x is :

Step-by-step solution

Plotting the diagram of the given function f ( x ) = max x 2 , 1 + [ x ] we get, Now when x ∈ [ 0 , 1 ) ⇒ f x = 1 When x ∈ [ 1 , 2 ) ⇒ f x = 2 And when x ∈ 2 , 2 ⇒ f x = x 2 So, from above diagram area can be calculated as, A = ∫ 0 1 1 . d x + ∫ 1 2 2 d x + ∫ 2 2 x 2 d x ⇒ A = 1 + 2 2 - 2 + 8 3 - 2 2 3 ⇒ A = 5 3 + 4 2 3