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Question: The value of ![](https://cdn.pureessence.tech/canvas_411.png?top_left_x=300&top_left_y=1116&width=30...

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

None of these

Explanation

Solution

For x Ī [0, 2], x2 + 1 Ī [1, 5],

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

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

Hence dx

= dx + 12x[x2+1]\int _ { 1 } ^ { \sqrt { 2 } } x ^ { \left[ x ^ { 2 } + 1 \right] } dx + 23x[x2+1]\int _ { \sqrt { 2 } } ^ { \sqrt { 3 } } x ^ { \left[ x ^ { 2 } + 1 \right] } dx + 32x[x2+1]\int _ { \sqrt { 3 } } ^ { 2 } x ^ { \left[ x ^ { 2 } + 1 \right] } dx

= 13\frac { 1 } { 3 }[23/2 – 1] + 15\frac { 1 } { 5 } 35/2.