`int{1+2tanx(tanx+secx)}^(1//2)dx` is equal to

Options

A. `log(secx+tanx)+C`
B. `log(secx+tanx)^(1//2)+C`
C. `log secx(secx+tanx)+C`
D. None of the above

Answer

C

Explanation

Let `l=int{1+2 tan x(tanx+secx)}^(1//2)dx`
`=int{1+2 tan^(2)x+2 tanx secx}^(1//2)dx`
`=int{sec^(2)x+tan^(2)x+2tanx secx}^(1//2)dx`
`=int(secx+tanx)dx`
`=log(secx+tanx)+log secx+C`
`=log secx(secx+tanx)+C`