Question

Given $\int e^x(\tan x+1)\sec xdx=e^{x}f\left(x\right)+c$. Find $f\left(x\right).$

Answer 1

It is given that $I=\int e^x\left(\tan x+1\right)\sec xdx=e^{x}f\left(x\right)+C.....\left(1\right)$

Consider

$I=\int e^x\left(\tan x+1\right)\sec xdx$

$=\int e^x\left(\sec x\tan x+\sec x\right)dx$

$=\int e^x\sec xdx+\int e^x\left(\sec x\tan x\right)dx$

$=\sec x\int e^{x}dx-\int e^x\left(\sec x\tan x\right)dx+\int e^x\left(\sec x\tan x\right)dx+C$

$=e^x\sec x+C........\left(2\right)$

Comparing equations (1) and (2) we get

$f\left(x\right)=\sec x$ .