# Integral $\int_{0}^{1}\ln x \, dx$

I have a question about the integral of $$\ln x$$.

When I try to calculate the integral of $$\ln x$$ from 0 to 1, I always get the following result.

• $$\int_0^1 \ln x = x(\ln x -1) |_0^1 = 1(\ln 1 -1) - 0 (\ln 0 -1)$$

Is the second part of the calculation indeterminate or 0?

What am I doing wrong?

Thanks Joachim G.

• How did you conclude that the second part is indeterminate? – Fabian Dec 25 '12 at 9:23

$\ln x$ is not defined at $0$ so the integral $$\int_0^1\ln x\, dx$$ is improper. Thus, $$\int_0^1\ln x\, dx=(x\ln x -x)|_0^1=1\ln 1-1-\lim_{x\to 0}x\ln x-x=-1+\lim_{x\to 0}x\ln x-x$$ We need to evaluate $$\lim_{x\to 0}x\ln x=\lim_{x\to 0}\frac{\ln x}{\frac1x}$$ Can you do that?
Looking sideways at the graph of $\log(x)$ you can also see that $$\int_0^1\log(x)dx = -\int_0^\infty e^{-x}dx = -1.$$
• +1 though I see $$- \int_{-\infty}^{0} e^x \, dx =-1$$ – Henry Dec 25 '12 at 10:05