Evaluate the following integrals:

Question:

Evaluate the following integrals:

$\int e^{x}[\sec x+\log (\sec x+\tan x)] d x$

Solution:

Let $I=\int e^{x}[\sec x+\log (\sec x+\tan x)] d x$

$I=\int e^{x} \sec x d x+\int \log (\sec x+\tan x) d x$

Integrating by parts

$=\int e^{x} \sec x d x+e^{x} \log (\sec x+\tan x)-\int e^{x} \sec x d x$

$=e^{x} \log (\sec x+\tan x)+c$

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