# Show that

Question:

$\int e^{x} \sec x(1+\tan x) d x$ equals

(A) $e^{x} \cos x+\mathrm{C}$

(B) $e^{x} \sec x+\mathrm{C}$

(C) $e^{x} \sin x+\mathrm{C}$

(D) $e^{x} \tan x+\mathrm{C}$

Solution:

$\int e^{x} \sec x(1+\tan x) d x$

Let $I=\int e^{x} \sec x(1+\tan x) d x=\int e^{x}(\sec x+\sec x \tan x) d x$

Also, let $\sec x=f(x) \Rightarrow \sec x \tan x=f^{\prime}(x)$

It is known that, $\int e^{x}\left\{f(x)+f^{\prime}(x)\right\} d x=e^{x} f(x)+\mathrm{C}$

$\therefore I=e^{x} \sec x+\mathrm{C}$

Hence, the correct answer is $\mathrm{B}$.