Question

The value of dx, where [x] is the greatest integer less than or equal to x is –

• A. 2
• B. 8/3
• C. 4
• D. None of these

Answer 1

Answer: (D)

None of these

Answer 2

For x Ī [0, 2], x² + 1 Ī [1, 5],

we must break [0, 2] = [0, 1] Ē [1, $\sqrt { 2 }$ ]

=[0, 1] Č [1, $\sqrt { 2 }$ ] Č[ $\sqrt { 2 }$ , $\sqrt { 3 }$ ] Č [ $\sqrt { 3 }$ , 2].

Hence dx

= dx + $\int _ { 1 } ^ { \sqrt { 2 } } x ^ { \left[ x ^ { 2 } + 1 \right] }$ dx + $\int _ { \sqrt { 2 } } ^ { \sqrt { 3 } } x ^ { \left[ x ^ { 2 } + 1 \right] }$ dx + $\int _ { \sqrt { 3 } } ^ { 2 } x ^ { \left[ x ^ { 2 } + 1 \right] }$ dx

= $\frac { 1 } { 3 }$[2^3/2 – 1] + $\frac { 1 } { 5 }$ 3^5/2.