If the solve the problem

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 $f(x)=1+\tan x$

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