The value of the integral $\int_{1}^{3}\left[x^{2}-2 x-2\right] d x$, where $[x]$ denotes the greatest integer less than or equal to $x$, is

JEE Main · 2021 · 24 Feb 2021 · Shift 2 · Definite Integrals

The value of the integral

\int_{1}^{3} [x^{2} - 2 x - 2] d x

, where

[x]

denotes the greatest integer less than or equal to

x

, is

Answer: (A)

[1, 3]

h (x) = x^{2} - 2 x - 2 = (x - 1)^{2} - 3

increases monotonically from

- 3

1

. It reaches integer levels at

h = - 2 (x = 2)

h = - 1 (x = 1 + 2)

h = 0 (x = 1 + 3)

. Hence

[h] = - 3

[1, 2)

- 2

[2, 1 + 2)

- 1

[1 + 2, 1 + 3)

0

[1 + 3, 3)

. So the integral

= - 3 (1) - 2 (2 - 1) - 1 (3 - 2) + 0 = - 1 - 2 - 3

, i.e. option (A).

JEE Main · 2021 · 24 Feb 2021 · Shift 2 · Single correct option

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