Giải bài 11 trang 152 SGK giải tích nâng cao 12
Cho biết \(\int\limits_{1}^{2}{f\left( x \right)dx=-4},\int\limits_{1}^{5}{f\left( x \right)dx=6},\)\(\int\limits_{1}^{5}g\left( x \right)dx=8.\). Hãy tính
a) \(\int\limits_{2}^{5}{f\left( x \right)dx}\); b) \(\int\limits_{1}^{2}{3f\left( x \right)dx}\);
c) \(\int\limits_{1}^{5}{\left[ f\left( x \right)-g\left( x \right) \right]dx}\) d) \(\int\limits_{1}^{5}{\left[ 4f\left( x \right)-g\left( x \right) \right]dx}\).
Ta có
\(\begin{aligned} a)\, & \int\limits_{2}^{5}{f\left( x \right)dx}=\int\limits_{1}^{5}{f\left( x \right)dx-\int\limits_{1}^{2}{f\left( x \right)dx=6+4}=10} \\ b)\, & \int\limits_{1}^{2}{3f\left( x \right)dx}=3\int\limits_{1}^{2}{f\left( x \right)dx}=3.\left( -4 \right)=-12 \\ c)\, & \int\limits_{1}^{5}{\left[ f\left( x \right)-g\left( x \right) \right]dx}=\int\limits_{1}^{5}{f\left( x \right)dx}-\int\limits_{1}^{5}{g\left( x \right)dx=6-8=-2} \\ d)\, & \int\limits_{1}^{5}{\left[ 4f\left( x \right)-g\left( x \right) \right]dx}=4\int\limits_{1}^{5}{f\left( x \right)dx}-\int\limits_{1}^{5}{g\left( x \right)dx=4.6-8=16} \\ \end{aligned} \)