It's on Indefinite Integrals

Question

$$\int \sqrt{ 1 + 2 \tan x ( \tan x + \sec x )} dx$$

Please tell me the way of solving such questions. like what could i assume sec x or sec x tan x to be equal to?

Answer 1

$\bf{My\; Solution::}$ Let $\displaystyle I=\int \sqrt{1+2\tan x \left(\tan x+\sec x\right)}dx = \int \sqrt{1+2\tan^2 (x)+2\tan x\cdot \sec x}dx$

So $\displaystyle I = \int \sqrt{1+\tan^2 x+\tan^2 x+2\tan x\cdot \sec x}dx=\int \sqrt{\left(\tan x+\sec x\right)^2}dx$

So $\displaystyle I = \int \tan xdx+\int \sec xdx = \ln\left|\sec x\right|+\ln \left|\sec x+\tan x\right|+\mathbb{C} = \ln \left|\sec x\cdot \left(\sec x+\tan x\right)\right|+\mathbb{C}$