Question

12．若$f（x）={x^3}+3\int{\begin{array}{l}1\\ 0\end{array}}f（x）dx$，则$\int{\begin{array}{l}1\\ 0\end{array}}f（x）dx$=-$\frac{1}{8}$．

Answer 1

分析 设$\int{\begin{array}{l}1\\ 0\end{array}}f（x）dx$=m，可得f（x）=x³+3m，因此m=$\int{\begin{array}{l}1\\ 0\end{array}}f（x）dx$=${∫}_{0}^{1}{x}^{3}dx$+3${∫}_{0}^{1}mdx$=$\frac{1}{4}{x}^{4}{|}_{0}^{1}$+3m=$\frac{1}{4}$+3m，可得m．

解答 解：设$\int{\begin{array}{l}1\\ 0\end{array}}f（x）dx$=m，
则f（x）=x³+3m，
∴m=$\int{\begin{array}{l}1\\ 0\end{array}}f（x）dx$=${∫}_{0}^{1}{x}^{3}dx$+3${∫}_{0}^{1}mdx$=$\frac{1}{4}{x}^{4}{|}_{0}^{1}$+3m=$\frac{1}{4}$+3m，
解得m=-$\frac{1}{8}$．
故答案为：$-\frac{1}{8}$．

点评 本题考查了微积分基本定理，考查了推理能力与计算能力，属于中档题．