Integration by parts of the logarithm

Problem

Compute $\displaystyle\int_1^e\ln x\,dx$ by parts.

Solution

We write $\ln x = 1\cdot\ln x$ and choose $f'=1 \to f=x$, $g=\ln x \to g'=\dfrac1x$: $\int_1^e\ln x\,dx = \bigl[x\ln x\bigr]_1^e - \int_1^e x\cdot\frac{1}{x}\,dx = (e\cdot 1 - 0) - [x]_1^e = e-(e-1) = \boxed{1}.$

Links

Topics: Integral
Concepts: Definite integral · Integration by parts
Methods: Integration by parts
Skills: Compute · Integrate
Exercise types: Integral computation