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Câu hỏi: Hàm số nào dưới đây là một nguyên hàm của hàm số $f\left( x \right)=x{{e}^{x}}$
A. $F\left( x \right)=\dfrac{{{x}^{2}}}{2}{{e}^{x}}$.
B. $F\left( x \right)=x{{e}^{x}}-{{e}^{x}}$.
C. $F\left( x \right)=x{{e}^{x}}+{{e}^{x}}$.
D. $F\left( x \right)=x{{e}^{x+1}}$.
A. $F\left( x \right)=\dfrac{{{x}^{2}}}{2}{{e}^{x}}$.
B. $F\left( x \right)=x{{e}^{x}}-{{e}^{x}}$.
C. $F\left( x \right)=x{{e}^{x}}+{{e}^{x}}$.
D. $F\left( x \right)=x{{e}^{x+1}}$.
Đặt $\left\{ \begin{aligned}
& u=x \\
& \text{dv}={{\text{e}}^{x}}\text{dx} \\
\end{aligned} \right.\Rightarrow \left\{ \begin{aligned}
& \text{du}=\text{dx} \\
& v={{\text{e}}^{x}} \\
\end{aligned} \right..$
Suy ra: $\int{x{{\text{e}}^{x}}\text{dx}}$ $=x{{\text{e}}^{x}}-\int{{{\text{e}}^{x}}\text{dx}}=x{{\text{e}}^{x}}-{{\text{e}}^{x}}+C$.
Vậy $\int{\text{x}{{\text{e}}^{x}}\text{dx}=}$ $x{{e}^{x}}-{{e}^{x}}$.
& u=x \\
& \text{dv}={{\text{e}}^{x}}\text{dx} \\
\end{aligned} \right.\Rightarrow \left\{ \begin{aligned}
& \text{du}=\text{dx} \\
& v={{\text{e}}^{x}} \\
\end{aligned} \right..$
Suy ra: $\int{x{{\text{e}}^{x}}\text{dx}}$ $=x{{\text{e}}^{x}}-\int{{{\text{e}}^{x}}\text{dx}}=x{{\text{e}}^{x}}-{{\text{e}}^{x}}+C$.
Vậy $\int{\text{x}{{\text{e}}^{x}}\text{dx}=}$ $x{{e}^{x}}-{{e}^{x}}$.
Đáp án B.