7.1 Integration By Parts/54

$y=5\ln(x),y=x\ln(x)$

${\begin{aligned}5\ln(x)&=x\ln(x)\\[1ex]&x=5\\[1ex]&x=1\\[1ex]5\ln(2)>2\ln(2)\end{aligned}}$

$\int _{1}^{5}\left(5\ln(x)-x\ln(x)\right)dx=\int _{1}^{5}\left(5\ln(x)\right)dx-\int _{1}^{5}\left(x\ln(x)\right)dx$

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) \right)dx = 5 \int_{1}^{5} \left(\ln(x) \right)dx }

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= 1 dx \\ du &= \frac{1}{x} dx \quad v=x \\ \end{align} }