Cho hàm số \(f\left( x \right)\) thỏa mãn \(\int\limits_0^3 {x.f\left( x \right).

* Hướng dẫn giải

Đặt \(\left\{ \begin{array}{l}
u = x\\
{\rm{d}}v = f'\left( x \right).{e^{f\left( x \right)}}{\rm{d}}x
\end{array} \right. \Rightarrow \left\{ \begin{array}{l}
{\rm{d}}u = {\rm{d}}x\\
v = {e^{f\left( x \right)}}
\end{array} \right..\) Khi đó \(\int\limits_0^3 {x.f'\left( x \right).{e^{f\left( x \right)}}{\rm{d}}x}  = x.{e^{f\left( x \right)}}\left| {\begin{array}{*{20}{c}}
3\\
0
\end{array}} \right. - \int\limits_0^3 {{e^{f\left( x \right)}}{\rm{d}}x} .\)

Suy ra \(8 = 3.{e^{f\left( 3 \right)}} - \int\limits_0^3 {{e^{f\left( x \right)}}{\rm{d}}x}  \to \int\limits_0^3 {{e^{f\left( x \right)}}{\rm{d}}x}  = 9 - 8 = 1.\)