The value of 9 ∫0⁹[√((10 x)/(x+1))] d x, where [t] denotes the greatest integer less than or equal to t, is

Solution

Maths sir · 3 steps

The inside only climbs from 0 to 3, so find where it crosses 1 and 2 and add up the rectangles.

1. Range of the inside. Let f(x) = √(10x/(x+1)). Since 10x/(x+1) = 10 - 10/(x+1), it increases with x. So f rises from f(0) = 0 to f(9) = √9 = 3, and [f(x)] can only be 0, 1 or 2 on [0, 9).
2. Find the jump points. f(x) < n means 10x/(x+1) < n², that is x < n²/(10 - n²). For n = 1, 2, 3 this gives x = 1/9, x = 2/3 and x = 9.

   Interval	[f(x)]	Length	Contribution
   0 ≤ x < 1/9	0	1/9	0
   1/9 ≤ x < 2/3	1	5/9	5/9
   2/3 ≤ x < 9	2	25/3	50/3

   ∫₀⁹ [√(10x/(x+1))] dx = 5/9 + 50/3 = 5/9 + 150/9 = 155/9

3. So. Multiplying by 9 gives 9 × 155/9 = 155.

The idea to keep. For the integral of [g(x)] with g monotonic, solve g(x) = n for each integer in its range. The integral is then a sum of rectangles: value times interval length.

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