# 做变限积分题的时候一定要摆脱思维定势

## 二、解析

$$k = x – t$$

$$\begin{cases} t = x – k \\ \mathrm{~d} t = – \mathrm{~d} k \\ x \in ( 0 , x ) \Rightarrow k \in ( x , 0 ) \end{cases}$$

\begin{aligned} I \\ & = \lim _ { x \rightarrow 0 } \frac { \int _ { 0 } ^ { \textcolor{springgreen}{x} } t f ( \textcolor{orangered}{x} – t ) \mathrm { d } t } { \textcolor{springgreen}{x} \int _ { 0 } ^ { \textcolor{springgreen}{x} } f ( \textcolor{orangered}{x} – t ) \mathrm { d } t } \\ \\ & = \lim_{x \rightarrow 0} \frac{- \int_{\textcolor{springgreen}{x}}^{0} (\textcolor{orangered}{x} – k) f(k) \mathrm{~d} k}{ – \textcolor{springgreen}{x} \int_{\textcolor{orangered}{x}}^{0} f(k) \mathrm{~d} k} \\ \\ & = \lim_{x \rightarrow 0} \frac{\int_{0}^{\textcolor{springgreen}{x}} [ \textcolor{springgreen}{x}f(k) – kf(k) ] \mathrm{~d} k }{\textcolor{springgreen}{x} \int_{0}^{\textcolor{springgreen}{x}} f(k) \mathrm{~d} k} \\ \\ & = \lim_{x \rightarrow 0} \frac{ \textcolor{springgreen}{x} \int_{0}^{\textcolor{springgreen}{x}} f(k) \mathrm{~d} k – \int_{0}^{\textcolor{springgreen}{x}} k f(k) \mathrm{~d} k }{\textcolor{springgreen}{x} \int_{0}^{\textcolor{springgreen}{x}} f(k) \mathrm{~d} k} \end{aligned}

\begin{aligned} I \\ \\ & \frac{0}{0} \Rightarrow \lim_{x \rightarrow 0} \frac{\int_{0}^{x} f(k) \mathrm{~d} k + xf(x) – xf(x)}{\int_{0}^{x} f(k) \mathrm{~d} k + xf(x)} \\ \\ & = \lim_{x \rightarrow 0} \frac{\int_{0}^{x} f(k) \mathrm{~d} k}{\int_{0}^{x} f(k) \mathrm{~d} k + xf(x)} \\ \\ & \frac{0}{0} \Rightarrow \lim_{x \rightarrow 0} \frac{f(x)}{f(x) + f(x) + xf^{\prime} (x)} \\ \\ & \frac{0}{0} \Rightarrow \lim_{x \rightarrow 0} \frac{f^{\prime}(x)}{f^{\prime}(x) + f^{\prime}(x) + f^{\prime}(x) + x f^{\prime \prime}(x) } \\ \\ & \frac{0}{0} \Rightarrow \lim_{x \rightarrow 0} \frac{f^{\prime \prime}(x)}{f^{\prime \prime}(x) + f^{\prime \prime}(x) + f^{\prime \prime} (x) + f^{\prime \prime}(x) + x \textcolor{red}{\boldsymbol{f^{\prime \prime \prime}(x)}} } \\ \\ & = \lim_{x \rightarrow 0} \frac{f^{\prime \prime}(x)}{f^{\prime \prime}(x) + f^{\prime \prime}(x) + f^{\prime \prime} (x) + f^{\prime \prime}(x)} \\ \\ & = \lim_{x \rightarrow 0} \frac{f^{\prime \prime}(x)}{4f^{\prime \prime}(x)} \\ \\ & = \frac{1}{4} \end{aligned}

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