Aloha :)
$$I=\int\limits_1^e\underbrace{x}_{=u'}\cdot\underbrace{\ln(x)}_{=v}\,dx=\left[\underbrace{\frac{x^2}{2}}_{=u}\cdot\underbrace{\ln(x)}_{=v}\right]_1^e-\int\limits_1^e\underbrace{\frac{x^2}{2}}_{=u}\cdot\underbrace{\frac1x}_{=v'}\,dx=\frac{e^2}{2}-\int\limits_1^e\frac x2\,dx$$$$\phantom I=\frac{e^2}{2}-\left[\frac{x^2}{4}\right]_1^e=\frac{e^2}{2}-\left(\frac{e^2}{4}-\frac14\right)=\frac{e^2+1}{4}$$
