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I have recently read a book on real analysis, more specific the part about Riemann Integrals. It says that if a function is bounded in a interval $[a,b]$, then it is integrable and also when a function has discontinuities such their measure is zero, then it is integrable. But the following examples are driving me crazy:

$$\lim_{t \to 0}\int_t^x \ln(s)ds\,\,\,, \,\,\, \lim_{t \to 0}\int_t^x \frac{ds}{s}$$

The left one is integrable because the discontinuity at zero has measure zero. Fine. But what about the right one? What happens? LET'S SUPPOSE WE DO NOT KNOW THEIR ANTIDERIVATIVES!

Thanks

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    $\begingroup$ Neither function is bounded on $[0,x]$ (nor are they even defined at $0$... but anyway, there's no way to extend it to a bounded function on $[0,x]$). However, for every $\alpha>0$ and $\beta > \alpha$, then the two functions you mention are continuous and bounded on $[\alpha, \beta]$, hence Riemann integrable on $[\alpha, \beta]$. $\endgroup$
    – peek-a-boo
    Commented May 29, 2020 at 2:19
  • $\begingroup$ I see, but why the left converges and the right does not. As is in the question, let's suppose we do not know their anti-derivatives. How could we be sure that they are integrable or not $\endgroup$
    – Mr. N
    Commented May 29, 2020 at 2:59

1 Answer 1

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Neither function is Riemann integrable. $\int_0^xln(s)ds=\lim_{t\to 0}\int_t^xln(s)ds=xln(x)-x$, where the latter is Riemann integrable.

$\int_0^x\frac{1}{s}ds=\lim_{t\to 0}\int_t^x\frac{1}{s}ds=\lim_{t\to 0}(ln(x)-ln(t))$ which is infinite.

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  • $\begingroup$ I meant that. I edited the question, take a look, please. I see, but what if I do not know its antiderivative, how should I proceed? Because when we find the antiderivative we just need to check if the limit exists. $\endgroup$
    – Mr. N
    Commented May 29, 2020 at 2:29
  • $\begingroup$ The answer I gave includes the anti-derivatives. If you don't know the anti-derivatives, you need to check what the Riemann sums look like, when you start at a small distance away from 0. $\endgroup$
    – herb steinberg
    Commented May 29, 2020 at 2:36
  • $\begingroup$ So I need to use the definition with summation?! Okay. Do you know any other references that I could take a look? Despite Wikipedia $\endgroup$
    – Mr. N
    Commented May 29, 2020 at 2:57
  • $\begingroup$ I assume you have a textbook discussing Riemann integration from the beginning. What you need should be there. My suggestion is to look at the piece of the sum near zero and see how fast it gets bigger as you go toward zero. $\endgroup$
    – herb steinberg
    Commented May 29, 2020 at 3:05

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