Integrate : $\sqrt { \bigg( \large\frac{1+ \sin 2x}{1+ \cos 2x}\bigg)}$$dx$

Question

\[\begin {array} {1 1} (a)\;\log(\sec x - \tan x)+c \\ (b)\;\{ \log (\sec x )+\tan x + c \\ (c)\;\{\log (\sec x)+x \} /\sqrt 2 +c \\ (d)\;\{ \log \sec x -x\} +c \end {array}\]

Answer 1

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.