If ∫ e ^ x ( x xf(x) + ( x x + se c ^ 2 x))dx = e ^ x f(x) + C,...

$∫ {e}^{sec x} ( sec xtan xf\left(x\right) + \left(sec xtan x + \sec^{2} x\right)dx = {e}^{sec x}f\left(x\right) + C$

Differentiating both sides w.r.t $‘x’$ we get

${e}^{\sec x}\left(\sec x \tan xf\left(x\right) + \left(\sec x \tan x + \sec^{2}x\right)\right) = \left({e}^{\sec x} ⋅ \sec x \tan x\right)f ( x ) + {e}^{\sec x}{f}^{'}\left(x\right)$

$⇒ {f}^{'}\left(x\right) = \sec^{2}x + \tan x \sec x$

$⇒ f\left(x\right) = ∫ \left(\sec^{2}x + \tan x \sec x\right)dx$

$⇒ f\left(x\right) = \tan x + \sec x + c, c \in R$
Hence, possible choice is $f\left(x\right) = \sec x + \tan x + \dfrac{1}{2} .$