If ∫ tan x 1 + tan x + tan 2 x d x = x - K A tan - 1 K tan x + 1 A + C , ( C is a constant of…

Let $I=\int \frac{\tan x}{1+\tan x+{\tan }^{2}x}dx$

$\Rightarrow I=\int \left(1-\frac{{\sec }^{2}x}{1+\tan x+{\tan }^{2}x}\right)dx$

$\Rightarrow I=x-\int \frac{dt}{1+t+t^2}$

Where, $\tan x=t\Rightarrow {\sec }^{2}xdx=dt$

$\therefore I=x-\frac{1}{\frac{\sqrt{3}}{2}}{\tan }^{-1}\left(\frac{t+\frac{1}{2}}{\frac{\sqrt{3}}{2}}\right)+C$, where $C$ is the constant of integration.

$=x-\frac{2}{\sqrt{3}}{\tan }^{-1}\left(\frac{2\tan x+1}{\sqrt{3}}\right)+C$

$\therefore K=2 \& A=3$.