Biết \(\int\limits_{1}^{2}{f\left( x \right)\text{d}x}=3\), \(\int\limits_{1}^{5}{f\left( x \right)\text{d}x}=4\). Tính \(\int\limits_{2}^{5}{\left( 2f\left( x \right)+x \right)\te...

* Hướng dẫn giải

Ta có \(\int\limits_{1}^{5}{f\left( x \right)\text{d}x}=4,\int\limits_{1}^{2}{f\left( x \right)\text{d}x}=3\Rightarrow \int\limits_{2}^{5}{f\left( x \right)\text{d}x}=\int\limits_{1}^{5}{f\left( x \right)\text{d}x}-\int\limits_{1}^{2}{f\left( x \right)\text{d}x}=1\).

\(\int\limits_{2}^{5}{\left( 2f\left( x \right)+x \right)\text{d}x}=2\int\limits_{2}^{5}{f\left( x \right)\text{d}x}+\int\limits_{2}^{5}{x\,\text{d}x}=2.1+\left. \frac{{{x}^{2}}}{2} \right|_{2}^{5}=\frac{25}{2}\).