The value of Σₙ₌₁¹⁰⁰ ∫ₙ₋₁ⁿ e^(x - [x])dx , where [ x ] is the greatest integer ≤ x, is :

Solution

Maths sir · 4 steps

On each interval from n − 1 to n, the fractional part restarts from 0, so all 100 pieces are equal.

1. Fractional part. x - [x] = {x} is the fractional part of x. On the interval n-1 ≤ x < n we have [x] = n-1, so the integrand is eˣ⁻⁽ⁿ⁻¹⁾.
2. One piece. Put t = x - (n-1). As x goes from n-1 to n, t goes from 0 to 1, so every piece becomes the same integral.

   ∫ₙ₋₁ⁿ e^(x-[x]) dx = ∫₀¹ eᵗ dt = e - 1

3. Add them up. Each of the 100 terms in the sum equals e - 1, so the total is 100 × (e-1).
4. So. The value is 100(e-1), which is option (b).

The idea to keep. The fractional part {x} has period 1. Integrate it over one unit interval from 0 to 1, then multiply by the number of unit intervals.

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