Integral of $\log(x)$ (or $\ln(x)$)

by Jakub Marian

Tip: See my list of the Most Common Mistakes in English. It will teach you how to avoid mis­takes with com­mas, pre­pos­i­tions, ir­reg­u­lar verbs, and much more (PDF Version).

The natural logarithm (of base $e$) is usually denoted $\log(x)$ in mathematics, but for the sake of students of other disciplines, we will denote it $\ln(x)$ in this article.

How to go about the integration? The task is actually very simple with the help of integration by parts, but it requires a little trick. As you can see, there is only one function in

$$ ∫ \ln(x)\,dx\,, $$

but integration by parts requires two. The trick is to write $\ln(x)$ as $1⋅\ln(x)$ and then apply integration by parts by integrating the $1$ and differentiating the logarithm:

$$ ∫ \ln(x)\,dx = ∫ \overbrace{1\vphantom{\ln(x)}}^{u’}\overbrace{\ln(x)}^v\,dx = \overbrace{x\vphantom{\ln(x)}}^u\overbrace{\ln(x)}^v - ∫ \overbrace{x\vphantom{\frac1x}}^u\overbrace{\frac1x}^{v’}\,dx = x\ln(x)-x+c\,. $$

By the way, I have written several educational ebooks. If you get a copy, you can learn new things and support this website at the same time—why don’t you check them out?

Enter the discussion 0