Question

$\int e^x\sec x(1+\tan x)dx$

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

Answer 1

The correct option is B $e^x\sec x+C$

Let $I=\int e^x\sec x(1+\tan x)dx=\int e^x(\sec x+\sec x\tan x)dx$
Also, let $\sec x=f\left(x\right)\Rightarrow \sec x\tan x=f^{\prime }\left(x\right)$
We know that, $\int e^x\left\{f\right(x)+f^{\prime }(x\left)\right\}=e^{x}f\left(x\right)+C$
$\therefore I=e^x\sec x+C$