Solve for: integral of (2)/(x * ln(x)^2) x

Expression: $\int{ \frac{ 2 }{ x \times {\ln\left({x}\right)}^{2} } } \mathrm{d} x$

Use the substitution $t=\ln\left({x}\right)$ to transform the integral

$\int{ \frac{ 2 }{ {t}^{2} } } \mathrm{d} t$

Use the property of integral $\begin{array} { l }\int{ a \times f\left( x \right) } \mathrm{d} x=a \times \int{ f\left( x \right) } \mathrm{d} x,& a \in ℝ\end{array}$

$2 \times \int{ \frac{ 1 }{ {t}^{2} } } \mathrm{d} t$

Use $\begin{array} { l }\int{ \frac{ 1 }{ {x}^{n} } } \mathrm{d} x=-\frac{ 1 }{ \left( n-1 \right) \times {x}^{n-1} },& n≠1\end{array}$ to evaluate the integral

$2 \times \left( -\frac{ 1 }{ t } \right)$

Substitute back $t=\ln\left({x}\right)$

$2 \times \left( -\frac{ 1 }{ \ln\left({x}\right) } \right)$

Simplify the expression

$-\frac{ 2 }{ \ln\left({x}\right) }$

Add the constant of integration $C \in ℝ$

$\begin{array} { l }-\frac{ 2 }{ \ln\left({x}\right) }+C,& C \in ℝ\end{array}$