If [ . ] represents the greatest integer function, then the value of
| ∫₀^(√(π/2)) [ [x²] - cos x ]dx | is ____________.

Solution

Maths sir · 3 steps

Split at x = 1, where [x²] jumps, and the integrand becomes a constant on each piece.

1. Pull out the integer. [x²] is always an integer, and [n + t] = n + [t] for any integer n. So the integrand is [x²] + [-cos x].
2. Where things change. The upper limit is √(π/2) ≈ 1.253, so x² runs from 0 to π/2 ≈ 1.571. That gives [x²] = 0 on [0, 1) and [x²] = 1 on [1, √(π/2)).

   Interval	[x²]	Range of -cos x	[-cos x]	Integrand
   [0, 1)	0	[-1, -0.54)	-1	-1
   [1, √(π/2))	1	[-0.54, -0.31)	-1	0

   ∫₀^(√(π/2)) [[x²] - cos x] dx = ∫₀¹ (-1) dx + ∫₁^(√(π/2)) 0 dx = -1

3. So. The integral is -1, and its absolute value is 1.

The idea to keep. With a greatest integer function inside an integral, find the points where the inside crosses an integer, split there, and integrate a constant on each piece.

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