Question

`int_0^4 x[x]  dx`, where [.] denotes the greatest integer function, equals ______

Options

• `13/2`

• 17

• `21/2`

• 19

Answer 1

`int_0^4 x[x]  dx`, where [.] denotes the greatest integer function, equals 17.

Explanation:

`int_0^4 x [x]  dx = int_0^1 x [x] dx + int_1^2 x [x] dx + int_2^3 x[x] dx + int_3^4 x[x] dx`

= `int_0^1 0 dx + int_1^2 x dx + int_2^3 2x dx + int_3^4 3x dx`

= `[x^2/2]_1^2 + [x^2]_2^3 + 3[x^2/2]_3^4`

= `3/2 + 5 + 21/2 = 17`