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Câu hỏi: Cho hàm số $f\left( x \right)=\left\{ \begin{aligned}
& 2x-1\text{ }khix>0 \\
& {{x}^{2}}-x-1khix\le 0 \\
\end{aligned} \right.. $ Tích phân $ \int\limits_{-3}^{2}{x.f'\left( 2x \right)dx}$ bằng
A. $\dfrac{50}{3}$
B. $\dfrac{13}{6}$
C. $\dfrac{19}{24}$
D. $\dfrac{11}{6}$
& 2x-1\text{ }khix>0 \\
& {{x}^{2}}-x-1khix\le 0 \\
\end{aligned} \right.. $ Tích phân $ \int\limits_{-3}^{2}{x.f'\left( 2x \right)dx}$ bằng
A. $\dfrac{50}{3}$
B. $\dfrac{13}{6}$
C. $\dfrac{19}{24}$
D. $\dfrac{11}{6}$
Cách giải:
Đặt $\left\{ \begin{aligned}
& u=x \\
& dv=f'\left( 2x \right)dx \\
\end{aligned} \right.\Rightarrow \left\{ \begin{aligned}
& du=dx \\
& v=\dfrac{1}{2}f\left( 2x \right) \\
\end{aligned} \right.$
Khi đó ta có
$I=\int\limits_{-2}^{2}{xf'\left( 2x \right)dx}=\dfrac{1}{2}xf\left( 2x \right)\left| \begin{aligned}
& 2 \\
& -2 \\
\end{aligned} \right.-\dfrac{1}{2}\int\limits_{-2}^{2}{f\left( 2x \right)dx}$
$=f\left( 4 \right)+f\left( -4 \right)-\dfrac{1}{4}\int\limits_{-2}^{2}{f\left( 2x \right)d\left( 2x \right)}$
$=f\left( 4 \right)+f\left( -4 \right)-\dfrac{1}{4}\int\limits_{-4}^{4}{f\left( x \right)dx}$
$=7+19-\dfrac{1}{4}\left( \int\limits_{-4}^{0}{\left( {{x}^{2}}-x-1 \right)dx}+\int\limits_{0}^{4}{\left( 2x-1 \right)dx} \right)$
$=26-\dfrac{1}{4}\left( \dfrac{76}{3}+12 \right)=\dfrac{50}{3}.$
Đặt $\left\{ \begin{aligned}
& u=x \\
& dv=f'\left( 2x \right)dx \\
\end{aligned} \right.\Rightarrow \left\{ \begin{aligned}
& du=dx \\
& v=\dfrac{1}{2}f\left( 2x \right) \\
\end{aligned} \right.$
Khi đó ta có
$I=\int\limits_{-2}^{2}{xf'\left( 2x \right)dx}=\dfrac{1}{2}xf\left( 2x \right)\left| \begin{aligned}
& 2 \\
& -2 \\
\end{aligned} \right.-\dfrac{1}{2}\int\limits_{-2}^{2}{f\left( 2x \right)dx}$
$=f\left( 4 \right)+f\left( -4 \right)-\dfrac{1}{4}\int\limits_{-2}^{2}{f\left( 2x \right)d\left( 2x \right)}$
$=f\left( 4 \right)+f\left( -4 \right)-\dfrac{1}{4}\int\limits_{-4}^{4}{f\left( x \right)dx}$
$=7+19-\dfrac{1}{4}\left( \int\limits_{-4}^{0}{\left( {{x}^{2}}-x-1 \right)dx}+\int\limits_{0}^{4}{\left( 2x-1 \right)dx} \right)$
$=26-\dfrac{1}{4}\left( \dfrac{76}{3}+12 \right)=\dfrac{50}{3}.$
Đáp án A.