Question

The value of `int_-1^3 (|x - 2| + [x])  dx` is equal to ______.

(where [.] denotes greatest integer function)

Options

• 3

• 5

• 4

• 7

Answer 1

The value of `int_-1^3 (|x - 2| + [x])  dx` is equal to 7.

(where [.] denotes greatest integer function)

Explanation:

Given, `int_-1^3 (|x - 2| + [x])  dx`

= `int_-1^0 {-(x - 2) - 1}  dx + int_0^1 -(x - 2) + 0  dx + int_1^2 {- (x - 2) + 1}  dx + int_2^3 (x - 2) + 2  dx`

= `int_1^0 (-x + 1)  dx + int_0^1 (-x + 2)  dx + int_1^2 (-x + 3)  dx + int_2^3 (x)  dx`

= `[(-x^2)/2 + x]_-1^0 + [(-x^2)/2 + 2x]_0^1 + [(-x^2)/2 + 3x]_1^2 + [x^2/2]_2^3`

= `3/2 + 3/2 + (-2 + 6) - ((-1)/2 + 3) + (9/2 - 4/2)`

= `3 + 4 - 5/2 + 5/2`

= 7