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 (syntax error): {\displaystyle \int_{1}^{5} \left(5\ln(x) \right)dx = 5 \int_{1}^{5} \left(\ln(x) \right)dx = 5\left(x\ln(x)- \int_{1}^{5} \left(\frac{x}{x} \right)dx \right) = 5\left(x\ln(x)- \int_{1}^{5} x |\bigg \right) }

${\begin{aligned}u&=\ln(x)\quad dv=1dx\\du&={\frac {1}{x}}dx\quad v=x\\\end{aligned}}$