Using trigonometry property:-
=> $\int \sqrt{ \large\frac{(\sin x + \cos x)^2}{2 \cos ^2 x}}$$dx$
=> $\large\frac{1}{\sqrt 2}$$ \int \large\frac{\sin x+ \cos x}{\cos^2 x }$$dx$
=> $\large\frac{1}{\sqrt 2}$$ \int (tan x +1) dx$
=> $\large\frac{1}{\sqrt 2} $$ \{ \log (\sec x)+ x \}+c$
Hence c is the correct answer.