Question
For any real number x , let x denotes the largest integer less than or equal to x . If I = ∫ 0 10 10 x x + 1 d x , then the value of 9 I is ____.
JEE Advanced
Mathematics
Definite Integration
2021
JEE Advanced 2021 (Paper 2)
JEE Advanced Mathematics Question (2021) — Solution
Options
- A. A
- B. B
- C. C
- D. D
Answer
A. A
Step-by-step solution
Given I = ∫ 0 10 10 x x + 1 d x Check the values of x at which the greatest integer function becomes integer. 10 x x + 1 = 1    ⇒    x = 1 9 10 x x + 1 = 4    ⇒    x = 2 3 10 x x + 1 = 9    ⇒    x = 9 I = ∫ 0 1 9 0 d x + ∫ 1 9 2 3 1 d x + ∫ 2 3 9 2 d x + ∫ 9 10 3 d x I = 0 + [ x ] 1 9 2 3 + [ 2 x ] 2 3 9 + [ 3 x ] 9 10 I = 2 3 - 1 9 + 18 - 4 3 + 30 - 27 I = 21 - 2 3 - 1 9 = 182 9 So, 9 I = 182
Related: Mathematics — Definite Integration · All PYQ Banks