If [x] denotes the greatest integer less than or equal to x, then the value of the integral ∫_(- π /2)^(π /2) [[x] - sin x]dx is equal to :

Solution

Maths sir · 3 steps

[x] is already an integer, so pull it out of the outer bracket and split the interval at every jump.

1. Pull out the integer. For any integer n, [n + y] = n + [y]. Here [x] is an integer, so [[x] - sin x] = [x] + [-sin x].
2. Find the jumps. [x] jumps at x = -1, 0, 1. On (-π/2, 0), -sin x lies in (0, 1), so [-sin x] = 0. On (0, π/2), -sin x lies in (-1, 0), so [-sin x] = -1. The endpoints are single points and do not change the integral.

   Interval	[x]	[-sin x]	Integrand	Length
   (-π/2, -1)	-2	0	-2	π/2 - 1
   (-1, 0)	-1	0	-1	1
   (0, 1)	0	-1	-1	1
   (1, π/2)	1	-1	0	π/2 - 1

   I = -2(π/2 - 1) + (-1)(1) + (-1)(1) + 0 = -π + 2 - 2 = -π

3. So. On each piece the integrand is a constant, so the integral is constant times length. The total is -π, option (a).

The idea to keep. Inside a greatest integer bracket, any integer part can be taken out: [n + y] = n + [y]. Then break the interval wherever any bracket jumps and add up constant times length.

Answer from the NTA final answer key. Solution by PrepWiser.