Question

The value of \(\int _2^4(\{x\}+[x])\,dx =\)...(where \(\{x\}\) and \([x]\) are the fractional part function and the greatest integer function, respectively)

• A. \(2\)
• B. \(4\)
• C. \(6\)
• D. \(8\)

Hint:

Since {x} + [x] = x, the integrand is simply x.

Answer 1

The Correct Option is C

Step 1: Key Identity:
Every real \(x\) satisfies \(x=[x]+\{x\}\). So \(\{x\}+[x]=x\).

Step 2: Integrate:
\[ \int_2^4x\,dx=\left[\frac{x^2}2\right]_2^4=\frac{16}2-\frac42=8-2=6 \]

Step 3: Check:
Direct split: on \([2,3)\) the integrand is \(x\), on \([3,4)\) it is also \(x\). The integral over both is the same 6. Option (A) 2 is just the interval length, (B) 4 and (D) 8 do not equal \(\int x\,dx\) on this interval.

Final Answer:
The value is 6, option (C). \[ \boxed{\text{(C) } 6} \]