Question:
If $\int e^{x}(\tan x+1) \sec x d x=e^{x} f(x)+C$, then write the value $f(x)$.
Solution:
Given, $\int e^{x}(\tan x+1) \sec x d x$
It is clearly of the form
$\int e^{x}\left[f(x)+f^{I}(x)\right] d x=e^{x} f(x)+c$
By comparison, $f(x)=1+\tan x ; f^{\prime}(x)=\sec x$
$=e^{x}(1+\tan x)+C$
Therefore, the value of $f(x)=1+\tan x$