Given :

Write f(x) satisfying the above.

∵ ^x (tan x + 1) Sec x dx = e^x f(x) + C

^x (Sec x tan x + Sec x) dx = e^x f(x) + C

⇒ ^x Sec x dx + ^x tan x Sec x dx = e^x f(x) + C

Integrating the first integral by parts:

⇒ e^x Sec x – ^x tan x Sec x dx + ^x tan x Sec x dx = e^x f(x) + C

⇒ e^x Sec x + C = e^x f(x) + C

On comparing we observe that -

f(x) = Sec x is the required value.