Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. It's 100% free, no registration required.

Sign up
Here's how it works:
  1. Anybody can ask a question
  2. Anybody can answer
  3. The best answers are voted up and rise to the top

How can one estimate the integral $$\int_e^x \log\log{t}\, dt$$ so that the error term is within $O\left(\frac{x}{\log^2x}\right)$? We may assume that $x>e$.

Any hint?

share|cite|improve this question
integration by part and bound logarithmic integral. – Soarer Sep 22 '11 at 7:28
@Michael, I'm curious, what does \nolimits do in the integral? The new version looks the same as the old to me. – Antonio Vargas Nov 25 '15 at 18:22
If the default value of limits will change (from current \nolimits to \limits), this integral will looks the same. – Michael Medvinsky Nov 25 '15 at 19:43
up vote 8 down vote accepted

If you integrate by parts, you end up with

$$x\log\log\,x-\int_e^x\frac{\mathrm dt}{\log\,t}$$

The last bit evaluates to $\mathrm{li}(x)-\mathrm{Ei}(1)$, where $\mathrm{Ei}(x)$ is the exponential integral, and $\mathrm{li}(x)=\mathrm{Ei}(\log\,x)$ is the logarithmic integral.

Using this asymptotic series for the exponential integral, we obtain


I've given the cookie, but it's up to you to supply the cream filling...

share|cite|improve this answer

Your Answer


By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.