7.1 Integration By Parts/28

$\int _{1}^{2}{\frac {\ln {x}^{2}}{x^{3}}}$

$u=\ln {x}\qquad dv={\frac {1}{x^{3}}}$

$du={\frac {1}{x}}\qquad v=-{\frac {1}{2x^{2}}}$

${\begin{aligned}\int _{1}^{2}{\frac {\ln {x}^{2}}{x^{3}}}&=-{\frac {\ln {x}}{2x^{2}}}-\int -{\frac {1}{2x^{3}}}\\[2ex]&=-{\frac {\ln {x}}{2x^{2}}}-{\frac {1}{2}}\int {\frac {1}{x^{3}}}\\[2ex]\end{aligned}}$