[m]\int _2^{\infty }\frac{dx}{x\ln ^2\left(x\right)}[/m]
[m] ∫ \frac{du}{u^2}= ∫ u^{-2}du=\frac{u^{-1}}{-1}=-\frac{1}{u}[/m]
[m]u=lnx[/m]
тогда
[m] ∫^{+ ∞}_{2} \frac{dx}{xln^2x}= lim_{A →+ ∞}∫^{A}_{2}\frac{dx}{xln^2x} =lim_{A →+ ∞}∫^{A}_{2}\frac{d(lnx)}{(lnx)^2} [/m]
[m]=lim_{A →+ ∞} (-\frac{1}{lnx})|^{A}_{2}=lim_{A → + ∞}(-\frac{1}{lnA})+\frac{1}{ln2}=0+\frac{1}{ln2}=\frac{1}{ln2} [/m]