`int {(1+2 tan x(tan x + sec x))^(1//2)` dx is equal to

Options

A. `log (sec x + tan x)+c`
B. `log (sec x + tanx)^(1//2) + c`
C. `log sec x (sec x + tanx) + c`
D. None of these

Answer

C

Explanation

Let
` I = int{1 +2 tan x (tan x + sec x)}^(1//2) dx`
` = int{(1+2 tan^(2) x + 2 tan x sec x}^(1//2) dx`
` = int{sec^(2) x + tan^(2) x + 2 tan x secx}^(1//2) dx`
`= int (sec x tan x) dx `
`log ( sec x + tan x) + log sec x + c`
`log sec x (sec x + tan x) + c`