Cho hàm số \(f\left( x \right)={{x}^{4}}-4{{x}^{3}}+2{{x}^{2}}-x+1\),\(\forall x\in \mathbb{R}\). Tính \(\int\limits_{0}^{1}{{{f}^{2}}\left( x \right).{f}'\left( x \right)\text{d}x}\). A.\(\frac{2}{3}\) B. \(2\) C. \(-\frac{2}{3}\). D. \(-2\).
Đáp án đúng: C Giải chi tiết:Ta có \(\int\limits_{0}^{1}{{{f}^{2}}\left( x \right).{f}'\left( x \right)\text{d}x}=\int\limits_{0}^{1}{{{f}^{2}}\left( x \right).\text{d}\left[ f\left( x \right) \right]}\)\(\left. =\frac{{{f}^{3}}\left( x \right)}{3} \right|_{0}^{1}\)\(=\frac{{{f}^{3}}\left( 1 \right)-{{f}^{3}}\left( 0 \right)}{3}\) \(=-\frac{2}{3}\). Chọn C