7.1 Integration By Parts/28

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \int_{1}^{2}\frac{(\ln{x})^2}{x^3} }

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle u = \ln^2{x} \qquad dv = \frac{1}{x^3} }

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle du = \frac{2\ln{(x)}}{x} \qquad v = -\frac{1}{2x^2} }

${\begin{aligned}\int _{1}^{2}{\frac {\ln {x}^{2}}{x^{3}}}=-{\frac {\ln ^{2}{(x)}}{2x^{2}}}-\int -{\frac {\ln {(x)}}{x^{3}}}=-{\frac {\ln ^{2}{(x)}}{2x^{2}}}+\int {\frac {\ln {(x)}}{x^{3}}}&=-{\frac {\ln ^{2}{(x)}}{2x^{2}}}-{\frac {\ln {(x)}}{2x^{2}}}-\int -{\frac {1}{2x^{3}}}\\[2ex]&u=\ln {(x)}\qquad dv={\frac {1}{x^{3}}}\\[2ex]&du={\frac {1}{x}}\qquad v=-{\frac {1}{2x^{2}}}\\[2ex]&=-{\frac {\ln ^{2}{(x)}}{2x^{2}}}-{\frac {\ln {(x)}}{2x^{2}}}-{\frac {1}{2}}\int {\frac {1}{x^{3}}}\\[2ex]&=-{\frac {\ln {(x)}}{2x^{2}}}-{\frac {1}{4x^{2}}}\end{aligned}}$