The value of `intsqrt(1+sec x)dx` is equal to

Options

A. `sin^(-1)(sqrt2 sin x)+C`
B. `2sin^(-1)(sqrt2 sin.(x)/(2))+C`
C. `2sin^(-1)(sqrt2 sin x)+C`
D. `2sin^(-1)((sqrt2x)/(2))+C`

Answer

B

Explanation

Let `l=int sqrt(1+secx)dx`
`=int(sqrt(1+cosx))/(sqrt(cosx))dx=int(sqrt2.cos.(x)/(2))/(sqrt(1-2 sin^(2).(x)/(2)))dx`
[let `t=sqrt2 sin.(x)/(2) rArr dt = (sqrt2)/(2). cos.(x)/(2).dx`]
`=int(2dt)/(sqrt(1-t^(2)))=2sin^(-1)(t)+C=2sin^(-1)(sqrt2. sin.(x)/(2))+C`