Đặt \(t=x^{2} \Rightarrow {\rm dt}=2x{\rm d}x.\)
Khi đó: \(\int x^{3} {\rm e}^{x^{2} } {\rm d}x =\frac{1}{2} \int t{\rm e}^{t} {\rm d}t.\)
Đặt \(\left\{\begin{array}{l} {u=t} \\ {{\rm d}v={\rm e}^{t} {\rm d}t} \end{array}\right. \Rightarrow \left\{\begin{array}{l} {{\rm d}u={\rm d}t} \\ {v={\rm e}^{t} } \end{array}\right. . \)
Vậy:\( \int x^{3} {\rm e}^{x^{2} } {\rm d}x =\frac{1}{2} \int t{\rm e}^{t} {\rm d}t=\frac{1}{2} t{\rm e}^{t} -\frac{1}{2} \int {\rm e}^{t} {\rm d}t =\frac{1}{2} t{\rm e}^{t} -\frac{1}{2} {\rm e}^{t} +C=\frac{1}{2} {\rm e}^{x^{2} } \left(x^{2} -1\right)+C.\)