Consider the following indefinite integral :
\begin{align} \int{{(1 + 2 \tan x(\tan x + \sec x)})^{1/2} dx} \end{align}
Clearly after simplifying the expression it would become :
\begin{align} \int{\sqrt{(\sec x + \tan x)^2} dx} \end{align}
And we know that
\begin{align} \sqrt{x^2} = |x| \end{align}
So, we can't directly write it as sec x + tan x (It can also be -(sec x + tan x) but I have seen in my book that the author just wrote it as like that and integrated it.
Why is it so?