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 $\mathrm{B}$.
Click here to get exam-ready with eSaral
For making your preparation journey smoother of JEE, NEET and Class 8 to 10, grab our app now.