Nguyên hàm 5

Tìm nguyên hàm : $\displaystyle\int\left(\dfrac{\ln x}{\ln x+2}\right)^2dx$

Ta có $$\begin{align*}\left(\dfrac{\ln x}{\ln x+2}\right)^2&=\dfrac{\ln^2 x}{\left(\ln x+2\right)^2}\\&=\dfrac{\ln^2 x+4\ln x+4-4\ln x-4}{\left(\ln x+2\right)^2}\\&=1-4\dfrac{\ln x+1}{\left(\ln x+2\right)^2} \\&=1-4\dfrac{\ln x+2-x\dfrac{1}{x}}{\left(\ln x+2\right)^2}\\&=1-4\dfrac{\left(\ln x+2\right)(x)'-x\left(\ln x+2\right)'}{\left(\ln x+2\right)^2}\\&=1-4\left(\dfrac{x}{\ln x+2} \right)'\end{align*}$$ Vậy  $\displaystyle\int\left(\dfrac{\ln x}{\ln x+2}\right)^2dx=x-4\dfrac{x}{2+\ln x}+C$