7.1 Integration By Parts/54
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 y={ \color{OliveGreen}5\ln(x) }, y={ \color{Red}x\ln(x)} }
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} 5\ln(x) &=x\ln(x)\\[1ex] &x=5 \\[1ex] &x=1 \\[1ex] \end{align} }
When 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 x=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 5\ln(2) > 2\ln(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 \int_{1}^{5}\left(5\ln(x) -x\ln(x) \right)dx = {\color{NavyBlue}\int_{1}^{5} \left(5\ln(x) \right)dx} - {\color{RedOrange}\int_{1}^{5} \left(x\ln(x) \right)dx } =25\ln(5)-20 - \left(\frac{25}{2}\ln(5) - 6 \right) = \frac{25}{2} \ln(5) -14 }
Failed to parse (Conversion error. Server ("https://en.wikipedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle {\begin{aligned}{\color {NavyBlue}\int _{1}^{5}\left(5\ln(x)\right)dx}&=5\int _{1}^{5}\left(\ln(x)\right)dx=5\left(x\ln(x){\bigg |}_{1}^{5}-\int _{1}^{5}\left({\frac {x}{x}}\right)dx\right)=5\left(x\ln(x){\bigg |}_{1}^{5}-x{\bigg |}_{1}^{5}\right)=5\left(5\ln(5)-1\ln(1)-\left(5-1\right)\right)=25\ln(5)-20\\[2ex]u&=\ln(x)\quad dv=1dx\\[2ex]du&={\frac {1}{x}}dx\quad v=x\\[2ex]\end{aligned}}}
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 {\color{RedOrange}\int_{1}^{5} \left(x\ln(x) \right)dx }= \frac{x^2\ln(x)}{2}\bigg|_{1}^{5} - \int_{1}^{5} \left(\frac{x^2}{2x} \right)dx = \frac{x^2\ln(x)}{2}\bigg|_{1}^{5} - \frac{1}{2}\int_{1}^{5} \left(x \right)dx = \frac{1\ln(1)}{2}-\frac{25\ln(5)}{2} -\left(\frac{1}{2}\right) \left( \frac{x^2}{2} \right) \bigg|_{1}^{5} = 0-\frac{25}{2}\ln(5) -\left(\frac{25-1}{4}\right) = \frac{25}{2}\ln(5) - 6 }
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} u &= \ln(x) \quad dv= x dx \\ du &= \frac{1}{x} \quad v=\frac{x^2}{2} \\ \end{align} }