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\int {\frac  {\ln x^{2}}{x}}\,dx\,

First, recognize that \ln x^{2}=2\ln x\,. So we have

\int {\frac  {\ln x^{2}}{x}}\,dx=2\int {\frac  {\ln x}{x}}\,dx\,

From here, we can do a substitution. Remember, the derivative of \ln x\, is {\frac  {1}{x}}\,.

u=\ln x\,

du={\frac  {1}{x}}dx\,

Substituting gives

2\int {\frac  {\ln x}{x}}\,dx=2\int u\,du=(u^{2})/2+C=((\ln x)^{2})/2)+C\,

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