Question

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

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

Answer 1

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


To find: $\int e^x\sec x(1+\tan x).dx$

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

Using $\int e^x\left[f\right(x)+f^{\prime }(x\left)\right].dx=e^{x}f\left(x\right)+C$

Here,

$f\left(x\right)=\sec x\Rightarrow f^{\prime }\left(x\right)=\sec x\tan x$

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

$=e^x\sec x+C$

$\therefore$ Option $\left(B\right)$ is correct.