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Câu hỏi: Cho hàm số $f\left( x \right)$ là hàm số chẵn và liên tục trên $\left[ -1;1 \right]$ thảo mãn $\int\limits_{-1}^{1}{f\left( x \right)\text{d}x}=\dfrac{86}{15},f\left( 1 \right)=5$. Khi đó $\int\limits_{0}^{1}{x}{f}'\left( x \right)\text{d}x$ bằng
A. $\dfrac{32}{15}$.
B. $\dfrac{86}{15}$.
C. $\dfrac{-11}{15}$.
D. $\dfrac{16}{15}$.
A. $\dfrac{32}{15}$.
B. $\dfrac{86}{15}$.
C. $\dfrac{-11}{15}$.
D. $\dfrac{16}{15}$.
Ta có hàm số $f\left( x \right)$ là hàm số chẵn và liên tục trên $\left[ -1;1 \right]$ suy ra $\int\limits_{-1}^{1}{f\left( x \right)\text{d}x}=\dfrac{86}{15}=2\int\limits_{0}^{1}{f\left( x \right)\text{d}x}\Rightarrow \int\limits_{0}^{1}{f\left( x \right)\text{d}x}=\dfrac{43}{15}$.
Xét $\int\limits_{0}^{1}{x}{f}'\left( x \right)\text{d}x$. Đặt $\left\{ \begin{aligned}
& u=x \\
& dv={f}'\left( x \right)\text{d}x \\
\end{aligned} \right.\Rightarrow \left\{ \begin{aligned}
& \text{d}u=\text{d}x \\
& v=f\left( x \right) \\
\end{aligned} \right.$.
Suy ra: $\int\limits_{0}^{1}{x}{f}'\left( x \right)\text{d}x=\left. x.f\left( x \right) \right|_{0}^{1}-\int\limits_{0}^{1}{f\left( x \right)\text{d}x}=5-\dfrac{43}{15}=\dfrac{32}{15}$.
Xét $\int\limits_{0}^{1}{x}{f}'\left( x \right)\text{d}x$. Đặt $\left\{ \begin{aligned}
& u=x \\
& dv={f}'\left( x \right)\text{d}x \\
\end{aligned} \right.\Rightarrow \left\{ \begin{aligned}
& \text{d}u=\text{d}x \\
& v=f\left( x \right) \\
\end{aligned} \right.$.
Suy ra: $\int\limits_{0}^{1}{x}{f}'\left( x \right)\text{d}x=\left. x.f\left( x \right) \right|_{0}^{1}-\int\limits_{0}^{1}{f\left( x \right)\text{d}x}=5-\dfrac{43}{15}=\dfrac{32}{15}$.
Đáp án A.