integrate : $\int \sec^3 x dx$

Question

\[\begin {array} {1 1} (a)\;\frac{1}{2}\{ \tan x \sec x - \log (\tan x - \sec x)\}+c \\ (b)\;\frac{1}{2}\{ \tan x \sec x + \log (\tan x + \sec x)\}+c \\ (c)\;\frac{1}{2} \{ \tan x \sec x + \log (\tan x - \sec x)\}+c \\ (d)\;None \end {array}\]

Answer 1

$ \int \sec x . \sec^2 x dx$

=> $ \int \sqrt{1+\tan ^2 x}. \sec^2. x dx$

$\tan x =t=> \sec^2 x dx=dt$

=> $\int \sqrt {1+t^2}.dt$

=> $\large\frac{1}{2}$$ t \sqrt {1+t^2} + \large\frac{1}{2}$$ \log \{ t+ \sqrt {1+t^2}\}+c$

=> $\large\frac{1}{2}$$ \tan x . \sec x +\large\frac{1}{2} $$ \log (\tan x + \sec x \}+c$

Hence b is the correct answer.