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$I=\int\limits_{\sqrt3}^{2\sqrt2}\frac14 \sqrt{x^2+1}.2lnx.2xdx=\int\limits_{\sqrt3}^{2\sqrt2} \frac14 \sqrt{1+x^2}lnx^2dx^2$Dat $t=\sqrt{x^2+1}\rightarrow t^2=x^2+1\rightarrow 2tdt=dx^2$doi can $x \sqrt3 2\sqrt2$ $t 2 3$$I=\frac12\int\limits_{2}^{3}t^2ln(t^2-1)dt$Dat $\begin{cases}u=ln(t^2-1) \\dv= t^2dt \end{cases}\rightarrow \begin{cases}dt=\frac{2t}{t^2-1}dt \\ v=\frac13t^3 \end{cases}$

$I=\int\limits_{\sqrt3}^{2\sqrt2}\frac14 \sqrt{x^2+1}.2lnx.2xdx=\int\limits_{\sqrt3}^{2\sqrt2} \frac14 \sqrt{1+x^2}lnx^2dx^2$Dat $t=\sqrt{x^2+1}\rightarrow t^2=x^2+1\rightarrow 2tdt=dx^2$doi can $x \sqrt3 2\sqrt2$ $t 2 3$$I=\frac12\int\limits_{2}^{3}t^2ln(t^2-1)dt$Dat $\begin{cases}u=ln(t^2-1) \\dv= t^2dt \end{cases}\rightarrow \begin{cases}dt=\frac{2t}{t^2-1}dt \\ v=\frac13t^3 \end{cases} .....$