Question

Integrala de la 1 la e din xlnx dx

Answer 1

$\int\limits^e_1 {x \ lnx} = \int\limits^e_1 {(\frac{x^2}{2})' \ lnx} = \frac{x^2}{2}lnx|^{e}_{1}- \int\limits^e_1 {\frac{x^2}{2} \ (lnx)' }$

$= \frac{e^2}{2} - \int\limits^e_1 {\frac{x^2}{2}* \frac{1}{x}} = \frac{e^2}{2}- \int\limits^e_1{\frac{x}{2}} = \frac{e^2}{2} - \frac{x^2}{4}|^{e}_{1}=\frac{e^2}{2}-\frac{e^2}{4}+\frac{1}{4}$

$= \frac{e^2+1}{4}$