∫13xdx\int \frac{1}{3x} dx

Standard integrals

Worked solution

1. 提取常数因数并应用标准积分公式求不定积分

根据常数倍积分性质与基本积分公式 ∫1x dx=ln⁡∣x∣+c\int \frac{1}{x} \, dx = \ln |x| + c：
∫13x dx=13∫1x dx=13ln⁡∣x∣+c\int \frac{1}{3x} \, dx = \frac{1}{3} \int \frac{1}{x} \, dx = \frac{1}{3} \ln |x| + c
（注：写作 13ln⁡(3x)+c′\frac{1}{3}\ln(3x) + c' 或 13ln⁡x+c\frac{1}{3}\ln x + c 亦同等正确）

Answer

13ln⁡∣x∣+c\frac{1}{3} \ln |x| + c

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