一个定积分可能存在多种形式的计算结果

一、题目

$$
I=\int \frac{2 \ln x-1}{x} \mathrm{~d} x = ?
$$

难度评级：

二、解析

解法一

$$
\begin{aligned}
I \\ \\
& = \int \frac{2 \ln x}{x} \mathrm{~d} x-\int \frac{1}{x} \mathrm{~d} x \\ \\
& = 2 \int \ln x \mathrm{~d} (\ln x)-\ln x + C_{1} \\ \\
& = \textcolor{springgreen}{ \ln ^{2} x-\ln x + C_{1} }
\end{aligned}
$$

或则：

$$
\begin{aligned}
I \\ \\
& = \int(2 \ln x-1) \mathrm{~d} (\ln x) \\ \\
& = 2 \int \ln x \mathrm{~d} (\ln x)-\int \mathrm{~d} (\ln x) \\ \\
& = \textcolor{springgreen}{ \ln ^{2} x-\ln x+C_{2} }
\end{aligned}
$$

解法二

$$
\begin{aligned}
I \\ \\
& = \int(2 \ln x-1) \mathrm{~d} (\ln x) \\ \\
& = \frac{1}{2} \int(2 \ln x-1) \mathrm{~d} (2 \ln x-1) \\ \\
& = \frac{1}{2} \cdot \frac{1}{2}(2 \ln x-1)^{2}+C_{3} \\ \\
& = \textcolor{springgreen}{ \frac{1}{4}(2 \ln x-1)^{2}+C_{3} }
\end{aligned}
$$

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