The smallest integral [a,b] such that
`int_(0)^(1) (1)sqrt(1+x^(4))dx in [a,b]` is given by

Options

A. `[1//sqrt(2),1]`
B. [0,1]
C. `[1//2,1]`
D. [3//4,1]`

Answer

A

Explanation

Let `I=overset(1)underset(0)int (1)/(sqrt(1+x^(4)))dx`.
We have,
`0 le x le 1 rArr0 le x^(4) le 1 rArr 1 le 1+x^(4) le 2`
`rArr1 le sqrt(1+x^(4))le sqrt(2)`
`rArr(1)/(sqrt(2))le(1)/(sqrt(1+x^(4))) le 1`
`(1)/(sqrt(2))(1-0) le overset(1)underset(0)int (1)/(sqrt(1+x^(4)))dx le 1(1-0)`
`(1)/(sqrt(2)) le 1 le 1 rArr I in [(1)/(sqrt(2)),1]`