[tex] \int\limits^e_1 {lnx} \, dx = \int\limits^e_1 {x'\ccdot lnx} \, dx = x\cdot lnx \Big|_1^e - \int\limits^e_1 {x\cdot (lnx)'} \, dx = \\ \\ =e\cdot lne - e\cdot ln1 - \int\limits^e_1 {x\cdot \dfrac{1}{x} } \, dx = e- 0 - \int\limits^e_1 {1} \, dx = \\ \\ =e - x\Big|_1^e = e-e+1 = 1[/tex]
