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Question: If \(x\) then the value of the integral \(\int _ { 0 } ^ { 2 } x ^ { 2 } [ x ] d x\) equals...

If xx then the value of the integral ∫02x2[x]dx\int _ { 0 } ^ { 2 } x ^ { 2 } [ x ] d x equals

A

5/3

B

7/3

C

8/3

D

4/3

Answer

7/3

Explanation

Solution

∫02x2[x]dx=∫01x2(0)dx+∫12x2(1)dx=0+[x33]12=73\int _ { 0 } ^ { 2 } x ^ { 2 } [ x ] d x = \int _ { 0 } ^ { 1 } x ^ { 2 } ( 0 ) d x + \int _ { 1 } ^ { 2 } x ^ { 2 } ( 1 ) d x = 0 + \left[ \frac { x ^ { 3 } } { 3 } \right] _ { 1 } ^ { 2 } = \frac { 7 } { 3 } .