Tìm nguyên hàm I = xe^3x dx.

Đặt $\left\{\begin{array}{c}\mathrm{u}=\mathrm{x}\\ \mathrm{dv}={\mathrm{e}}^{3\mathrm{x}}\mathrm{dx}\end{array}\right.$ Û $\left\{\begin{array}{c}\mathrm{du}=\mathrm{dx}\\ \mathrm{v}=\frac{1}{3}{\mathrm{e}}^{3\mathrm{x}}\end{array}\right.$ 

Do đó:  I =$\int {\mathrm{xe}}^{3\mathrm{x}}\mathrm{dx}=\frac{{\displaystyle 1}}{{\displaystyle 3}}{\mathrm{xe}}^{3\mathrm{x}}-\frac{{\displaystyle 1}}{{\displaystyle 3}}\int {\mathrm{e}}^{3\mathrm{x}}\mathrm{dx}$  

= $\frac{1}{3}\mathrm{x}{\text{e}}^{3\mathrm{x}}-\frac{1}{9}{\mathrm{e}}^{3\mathrm{x}}+\mathrm{C}$.