第233题 | 知识点：强化极限训练（二十三）

题目

$$
\lim_{x\to0}
\int_0^x\Big(\frac{\arctan t}{t}\Big)^{\dfrac{1}{\int_0^t\ln(1+u)du}}\cot x dt
$$

解答

$$
\begin{aligned}
&
\lim_{x\to0}
\int_0^x\Big(\frac{\arctan t}{t}\Big)^{\dfrac{1}{\int_0^t\ln(1+u)du}}\cot x dt
\\
=&
\lim_{x\to0}
\dfrac{\displaystyle\int_0^x \Big(\dfrac{\arctan t}{t}\Big)^{\dfrac{1}{\int_0^t\ln(1+u)du}}dt}{\tan x}
\\
=&
\lim_{x\to0}
\Big(\dfrac{\arctan t}{t}\Big)^{\dfrac{1}{\int_0^t\ln(1+u)du}}dt
\\
&
\lim_{x\to0}
\dfrac{\ln(\dfrac{\arctan t}{t})}{\displaystyle\int_0^t \ln(1+u)du}
\\
=&
\lim_{x\to0}
\dfrac{\arctan t - t}{t\displaystyle\int_0^t udu}
\\
=&
-\dfrac{2}{3}
\\
&
\lim_{x\to0}
\int_0^x\Big(\frac{\arctan t}{t}\Big)^{\dfrac{1}{\int_0^t\ln(1+u)du}}\cot x dt
\\
=& e^{-\frac{2}{3}}
\end{aligned}
$$