Let [t] denote the greatest integer function. If ∫0^(2.4)[x²] d x=α+β √2+γ √3+δ √5, then α+β+γ+δ is equal to __________.

Solution

Maths sir · 3 steps

Cut the interval wherever x² hits an integer, at x = √n, then add up the rectangles.

1. Find the breakpoints. [x²] changes value when x² is an integer, that is at x = 1, √2, √3, 2, √5. The next one, √6 ≈ 2.449, lies beyond 2.4. Since 2.4² = 5.76, the last piece has [x²] = 5.

   Interval	[x²]	Area of the piece
   0 to 1	0	0
   1 to √2	1	√2 - 1
   √2 to √3	2	2√3 - 2√2
   √3 to 2	3	6 - 3√3
   2 to √5	4	4√5 - 8
   √5 to 2.4	5	12 - 5√5

2. Add the pieces. Constants: -1 + 6 - 8 + 12 = 9. Coefficient of √2: 1 - 2 = -1. Coefficient of √3: 2 - 3 = -1. Coefficient of √5: 4 - 5 = -1.

   ∫₀^(2.4)[x²] dx = 9 - √2 - √3 - √5

3. So. α = 9 and β = γ = δ = -1, so α + β + γ + δ = 9 - 3 = 6.

The common slip: -1. Stopping at √5 and missing the last piece, where [x²] = 5 because 2.4² = 5.76, gives -1 instead of 6.

The idea to keep. For [f(x)] inside an integral, find where f(x) hits each integer, split there, and add value times width. Always check which integer the upper limit falls under.

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