Cho hàm số f(x) thỏa mãn f'(x)=x e^x và f(0)=2 Tính f(1)

* Hướng dẫn giải

Đáp án D.

$$\begin{gathered} \mathrm{I}={\int }_0^1\mathrm{f}\text{'}\left(\mathrm{x}\right)d\mathrm{x}={\int }_0^1{\mathrm{xe}}^{\mathrm{x}}d\mathrm{x}=\mathrm{f}\left(1\right)-\mathrm{f}\left(0\right) \\ \mathrm{Đặt} \left\{\begin{array}{l}\mathrm{x}=\mathrm{u}\\ {\mathrm{e}}^{\mathrm{x}}\mathrm{dx}=\mathrm{dv}\end{array}\iff \left\{\begin{array}{l}\mathrm{dx}=\mathrm{du}\\ \mathrm{v}={\mathrm{e}}^{\mathrm{x}}\end{array}\right.\right. \\ \Rightarrow \mathrm{I}={\overline{){\mathrm{xe}}^{\mathrm{x}}}}_0^1-{\int }_0^1{\mathrm{e}}^{\mathrm{x}}d\mathrm{x}=\mathrm{e}-{\overline{){\mathrm{e}}^{\mathrm{x}}}}_0^1=\mathrm{e}-(\mathrm{e}-1)=1 \\ \Rightarrow \mathrm{f}\left(1\right)=\mathrm{f}\left(0\right)+\mathrm{I}=2+1=3 \end{gathered}$$