Let `I=underset(0) overset(1) int(sinx)/(sqrtx) dx and f= underset(o)overset(1) int( cos x)/(sqrtx) dx`
Then , which one of the following is true ?

Options

A. `I gt (2)/(3)" and "J gt 2`
B. `I lt (2)/(3)" and "J lt 2`
C. `I lt (2)/(3)" and "J gt 2`
D. `I gt (2)/(3)" and "J lt 2`

Answer

B

Explanation

We have,
`sin x lt x` for `0 lt x lt 1`
`rArr (sin x)/(sqrt(x)) lt (x)/(sqrt(x))" for "0 lt x lt 1`
`rArr overset(1)underset(0)int (cos)/(sqrt(x))dx lt overset(1)underset(0)int sqrt(x)dx rArr I lt (2)/(3)[x^(3//2)]_(0)^(1) rArrI lt (2)/(3)`
Also,
`m le f(x) le M` for all `x in (0,1)`
`rArr (cos x)/(sqrt(x)) lt (1)/(sqrt(x))dx rArrJ lt [x^(1//2)]_(0)^(1) rArrJ lt 2`.