∫1/x dx = log|x| + C

The power formula for $x^n$ fails at $n=-1$ because the denominator becomes $0$. $\ln|x|$ fills that gap.

The absolute value is there so the formula also works for negative $x$.

Examples

1. 01Integrate$\displaystyle \int \frac{1}{x} \, dx$AnswerHide

   Answer$\displaystyle \log_e|x| + C$
2. 02Find the value$\displaystyle \int_{1}^{4} \frac{1}{x^2} \, dx$AnswerHide

   Answer$\displaystyle \frac{3}{4}$
3. 03For which n does this formula fail?$\displaystyle \int x^{n}\,dx = \frac{x^{n+1}}{n+1}$AnswerHide

   Answer$\displaystyle n=-1$
4. 04Find the value$\displaystyle \int_{3}^{5} \frac{1}{x^2} \, dx$AnswerHide

   Answer$\displaystyle \frac{2}{15}$
5. 05Find the value$\displaystyle \int_{4}^{6} \frac{1}{x^2} \, dx$AnswerHide

   Answer$\displaystyle \frac{1}{12}$
6. 06What is this?$\displaystyle \int \frac{2}{x}\,dx$AnswerHide

   Answer$\displaystyle 2\log_e|x|+C$

FAQ

Q1Why the absolute value?Show answerHide

A

So that $\frac{1}{x}$ has an antiderivative for $x<0$ too. Differentiating $\log|x|$ gives $\frac{1}{x}$ regardless of the sign of x.

Practice this pattern in the app, 5 problems a day.

Published 2026-09-03 · Updated 2026-09-05