Nếu \(\int\limits_{0}^{\frac{\pi }{3}}{\left[ \sin x-3f\left( x \right) \right]}\text{d}x=6\) thì \(\int\limits_{0}^{\frac{\pi }{3}}{f\left( x \right)}\text{d}x\) bằng

* Hướng dẫn giải

Ta có \(6=\int\limits_{0}^{\frac{\pi }{3}}{\left[ \sin x-3f\left( x \right) \right]}\text{d}x=\int\limits_{0}^{\frac{\pi }{3}}{\sin x\text{d}x-3\int\limits_{0}^{\frac{\pi }{3}}{f\left( x \right)}}\text{d}x=-\left. \text{co} \text{s}\,x \right|_{0}^{\frac{\pi }{3}}-3\int\limits_{0}^{\frac{\pi }{3}}{f\left( x \right)}\text{d}x=\frac{1}{2}-3\int\limits_{0}^{\frac{\pi }{3}}{f\left( x \right)}\text{d}x\)

Suy ra \(3\int\limits_{0}^{\frac{\pi }{3}}{f\left( x \right)}\text{d}x=\frac{1}{2}-6\Leftrightarrow \int\limits_{0}^{\frac{\pi }{3}}{f\left( x \right)}\text{d}x=-\frac{11}{6}\).