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} }

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

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 \begin{align} \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^2{(x)}}{2x^2} -\frac{\ln{(x)}}{2x^2} - \frac{1}{4x^2} \bigg|_{1}^{2} \\[2ex] & = -\frac{\ln^2{(2)}}{8} -\frac{\ln{(2)}}{8} - \frac{3}{16} \\[2ex] \end{align} }