$$\int \frac{1}{3x} dx$$

1. How do I solve this?
2. <details class="reasoning-details"> <summary style="cursor:pointer; color:#666; font-size:0.9em;">Reasoning (click to expand)</summary> <div class="reasoning-block"> **Evaluating Integrals** I'm reviewing the student's work on the integral $\int \frac{1}{3x} \, dx$. They've offered two distinct approaches, and I'm currently assessing the validity of both solutions, specifically whether each is considered "correct" in its entirety. </div> </details> Yes, **both are correct!** They look different at first glance, but try using a logarithm property to expand the first one: $$\frac{1}{3} \log|3x| = \frac{1}{3}\big(\log 3 + \log|x|\big)$$ What happens when you distribute the $\frac{1}{3}$ and combine the constant term with $+ C$?
3. 这道题怎么做？
4. <details class="reasoning-details"> <summary style="cursor:pointer; color:#666; font-size:0.9em;">思考过程 (点击展开)</summary> <div class="reasoning-block"> **Analyzing Student Inquiry** I'm reviewing a student's question about solving an integral. They've presented two approaches, one using direct u-substitution or chain rule, resulting in $\frac{1}{3} \log |3x| + C$, and another that starts by factoring out a constant. I'm comparing these methods for clarity and correctness. </div> </details> **两个都完全正确！** 你可能好奇为什么它们长得不一样却都是对的。你可以用对数的性质把左边的式子展开看一看： $$\frac{1}{3}\log|3x| = \frac{1}{3}\big(\log 3 + \log|x|\big) = \frac{1}{3}\log|x| + \frac{1}{3}\log 3$$ 式子中的 $\frac{1}{3}\log 3$ 本身是一个常数，它可以直接并入任意常数 $+ C$ 里面（变成一个新的常数 $C'$）。 所以两者在数学上是等价的！平时做题时，**右边的方法（先提常数因数）**更推荐，计算起来更不容易漏掉系数。