If $\int e^{\sec x}\left(\sec x \tan x f(x)+\sec x \tan x+\sec ^2 x\right) d x \quad=e^{\sec x} f(x)+C$, then a possible choice o $f(x)$ is :
Select the correct option:
A
$\sec x-\tan x-\frac{1}{2}$
B
$\sec x+\tan x+\frac{1}{2}$
C
$\sec x+\tan x-\frac{1}{2}$
D
$x \sec x+\tan x+\frac{1}{2}$
✓ Correct! Well done.
✗ Incorrect. Try again or view the solution.
Solution
$$
\begin{aligned}
\int e^{\sec x}(\sec x \tan x f(x)+(\operatorname{sex} x \tan x & \left.\left.+3 e^2 x\right)\right) d x \\
& =e^{\sec x} f(x)+C
\end{aligned}
$$
$\therefore \quad$ We know that
$$
\begin{array}{ll}
\int e^{g(x)} & \left(\left(g^{\prime}(x) f(x)\right)+f^{\prime}(x)\right) d x=e^{g(x)} \times f(x)+C \\
\therefore & f(x)=\int\left((\sec x \tan x)+\sec ^2 x\right) d x \\
\therefore & f(x)=\sec x+\tan x+C
\end{array}
$$
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