Question

Evaluate : $\int _ { - 2 } ^ { 2 } \max${x + |x|, x – [x]} dx, where [x] denotes the greatest integer £ x –

• A. 4
• B. 3
• C. 5
• D. 1

Answer 1

Answer: (C)

5

Answer 2

$\int _ { - 2 } ^ { 2 } \max${x + |x|, x – [x]} dx

= $\int _ { - 2 } ^ { 0 } \max$ {0, x – [x]} dx + $\int _ { 0 } ^ { 2 } \max$ {2x, x – [x]}dx

the graph of ƒ(x) = max {x + |x|, x – [x]} is shown as in figure

therefore $\int _ { - 2 } ^ { 2 } \max${x + |x|, x – [x]} dx = Area of shaded region

= 2 $\left( \frac { 1 } { 2 } \times 1 \times 1 \right)$ + $\frac { 1 } { 2 }$ × 2 × 4

= 1 + 4

= 5.