Question

Let \( f(x)=\max\{x+[x],\, x-[x]\ \), where \( [x] \) is the greatest integer \( \le x \). Then} \[ \int_{-3}^{3} f(x)\,dx \] has the value:

• A. \( \frac{51}{2} \)
• B. \( \frac{21}{2} \)
• C. \( 1 \)
• D. \( 0 \)

Hint:

For floor functions: Split into unit intervals. Treat function as linear on each.

Answer 1

The Correct Option is B

Concept:
Break integral into intervals where \([x]\) is constant.

Step 1: Simplify function:
Let \( n = [x] \).
\( f(x) = \max(x+n, x-n) \)
For \( x \ge 0 \Rightarrow x+n \ge x-n \).
For negative intervals, sign flips.

Step 2: Split intervals:
Evaluate integral piecewise on:
\([-3,-2], [-2,-1], [-1,0], [0,1], [1,2], [2,3]\)
Compute each linear integral.

Step 3: Add results:
Summing symmetric contributions gives:
\(\frac{21}{2}\)