$\int tan x(\sec x - \tan x) dx =$

sec x - tan x + x - C sec x - tan2x + x - C sec x + tan x + x + C sec x - tan x + C

sec x - tan x + x - C

$\int tan x(\sec x - \tan x) dx$ $\int \sec x \tan x dx - \int \tan^2 x dx$ as 1 + tan2x = sec2x so sec2x - 1 = tan2x  = $\sec x - \int (\sec^2x - 1) dx$ so sec x - tan x + x - C