Question

$\int e^x\sec x(1+\tan x)dx$ is equal to
(where $c$ is integration constant)

• A. $e^x\sec x+c$
• B. $e^x\tan x+c$
• C. $e^x(\sec x+\tan x)+c$
• D. $e^x(\sec x-\tan x)+c$

Answer 1

The correct option is A $e^x\sec x+c$

$I=\int e^x\sec x(1+\tan x)dx=\int e^x(\sec x+\sec x\tan x)dx\underset{f\left(x\right)}{\downarrow }\underset{f^{\prime }\left(x\right)}{\downarrow }\Rightarrow I=e^x\sec x+c[\because \int e^x\left(f\right(x)+f^{\prime }(x\left)\right)dx=e^{x}f\left(x\right)+c]$