Câu hỏi: Nếu $\int_{-1}^{2}{f\left( x \right)dx}=2$ và $\int_{-1}^{2}{g\left( x \right)dx=-1}$ thì $\int_{-1}^{2}{\left[ x+2f\left( x \right)-3g\left( x \right) \right]}dx$ bằng
A. $\dfrac{11}{2}.$
B. $\dfrac{17}{2}.$
C. $\dfrac{7}{2}.$
D. $\dfrac{5}{2}.$
A. $\dfrac{11}{2}.$
B. $\dfrac{17}{2}.$
C. $\dfrac{7}{2}.$
D. $\dfrac{5}{2}.$
Ta có:
$\begin{aligned}
& I=\int_{-1}^{2}{\left[ x+2f\left( x \right)-3g\left( x \right) \right]}dx =\int_{-1}^{2}{xdx}+2\int_{-1}^{2}{f\left( x \right)dx}-3\int_{-1}^{2}{g\left( x \right)dx} \\
& =\left. \dfrac{{{x}^{2}}}{2} \right|_{-1}^{2}+2.2-3.\left( -1 \right)=\dfrac{3}{2}+4+3=\dfrac{17}{2}. \\
\end{aligned}$
$\begin{aligned}
& I=\int_{-1}^{2}{\left[ x+2f\left( x \right)-3g\left( x \right) \right]}dx =\int_{-1}^{2}{xdx}+2\int_{-1}^{2}{f\left( x \right)dx}-3\int_{-1}^{2}{g\left( x \right)dx} \\
& =\left. \dfrac{{{x}^{2}}}{2} \right|_{-1}^{2}+2.2-3.\left( -1 \right)=\dfrac{3}{2}+4+3=\dfrac{17}{2}. \\
\end{aligned}$
Đáp án B.