If e^ x( x x,f(x)+( x x+^2x))dx=e^ xf(x)+C, then a possible choice of f(x) is:

Question

If $\displaystyle\int e^{\sec x}\left(\sec x	an x\,f(x)+(\sec x	an x+\sec^2x)ight)dx=e^{\sec x}f(x)+C$, then a possible choice of $f(x)$ is:

Accepted Answer

Using $\displaystyle\int e^{g(x)}\left(g'(x)f(x)+f'(x)\right)dx=e^{g(x)}f(x)+C$ with $g(x)=\sec x$ (so $g'(x)=\sec x\tan x$), comparing $f'(x)=\sec x\tan x+\sec^2x$:
$$f(x)=\int(\sec x\tan x+\sec^2x)\,dx=\sec x+\tan x+C$$

Taking $C=\dfrac12$: $f(x)=\sec x+\tan x+\dfrac12$

Solutions · 1