Ví dụ: Tính tích phân $I = \int\limits_1^2 {\ln tdt} .$

Giải: Đặt $\left\{ \begin{array}{l}u = \ln t\\dv = dt\end{array} \right. \Rightarrow \left\{ \begin{array}{l}du = \dfrac{{dt}}{t}\\v = t\end{array} \right.$.

Khi đó $I = t\ln t\left| {_{\scriptstyle\atop\scriptstyle1}^{\scriptstyle2\atop\scriptstyle}} \right. - \int\limits_1^2 {dt}  = t\ln t\left| {_{\scriptstyle\atop\scriptstyle1}^{\scriptstyle2\atop\scriptstyle}} \right. - t\left| {_{\scriptstyle\atop\scriptstyle1}^{\scriptstyle2\atop\scriptstyle}} \right. = 2\ln 2 - 1.$