The value of the definite integral `int_0^(pi/2)sqrt(tanx)dx` is `sqrt(2)pi` (b) `pi/(sqrt(2))` `2sqrt(2)pi` (d) `pi/(2sqrt(2))`

Options

A. `sqrt(2)pi`
B. `(pi)/(sqrt(2))`
C. `2sqrt(2)pi`
D. `(pi)/(2sqrt(2))`

Answer

B

Explanation

`I=int_(0)^(pi//2) sqrt(tanx) dx`…………….1
or `I=int_(0)^(pi//2) sqrt(cotx)dx`…………….2
Adding 1 and 2 we get
`2I=int_(0)^(pi//2) (sqrt(tanx)+sqrt(cotx))dx`
`=sqrt(2)int_(0)^(pi//2) (sinx+cosx)/(sqrt(sin2x)) dx`
`=sqrt(2) int_(0)^(pi//2) (sinx+cosx)/(sqrt(1-(sinx-cosx)^(2)))dx`
`=sqrt(2)int_(-1)^(1)(dt)/(sqrt(1-t^(2)))` (Putting `sinx-cosx=1`)
`=2sqrt(2)int_(0)^(1)(dt)/(sqrt(1-t^(2)))`
`=2sqrt(2)pi`
or `I=(pi)/(sqrt(2))`