Question

$\int e^{\tan x}(\sec x-\sin x)dx$ is equal to

• A. $e^{\tan x}\cos x+C$
• B. $e^{\tan x}\sin x+C$
• C. $-e^{\tan x}\cos x+C$
• D. $e^{\tan x}\sec x+C$

Answer 1

The correct option is A $e^{\tan x}\cos x+C$

$I=\int e^{\tan x}(\sec x-\sin x)dx$
$=-\int \sin x{e}^{\tan x}dx+\int \sec x{e}^{\tan x}dx$
$=e^{\tan x}\cos x-\int \cos x{e}^{\tan x}{\sec }^{2}xdx+\int \sec x{e}^{\tan x}dx$
$=\cos x{e}^{\tan x}+C$