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: Geometry:
The graph of $y=x$ between $x=2$ and $x=4$ forms a trapezoid with parallel sides 2 and 4 and width 2.

Step 2: Area:
Area $=\dfrac{2+4}{2}\times2=6$.

Step 3: Answer:
Option (C).

Final Answer:
Option (C). \[ \boxed{\text{(C) } 6} \]