Some Irresistible Integrals, Computed Using Statistical Concepts
Below are a few integrals that you won’t find in textbooks. Solving them is a good exercise for college students with some advanced calculus training. We provide the solution, as well as a general framework to compute many similar integrals. Maybe this material should be part of the standard math curriculum. Here, p, q, r are positive real numbers, with q larger than p.
The Gamma symbol represents the gamma function. It is possible that these results are published here for the first time. These are known as Frullani integrals, although the ones mentioned here are not covered by Frullani’s theorem, nor any recent generalization that I am aware of. Indeed, AI-based automated integration platforms such as WolframAlpha can not find the exact value (only an approximation) while they are able to compute standard Frullani integrals exactly. My approach to derive the exact values is different from the classical approaches, as it relies on the statistical concept of expectation, possibly leading to interesting areas of research.
How to compute such integrals?
These integrals are a particular case of the following main result, proved in the next section:
where g(x) / x tends to 1 as x tends to infinity, and f is a bounded function with a finite expectation. Some additional conditions may be required, for instance the fact that there is no singularity point in the above quotient, and that g(x) has a lower bound that is strictly positive. The expectation of f, also called average value, is defined as
For instance, if f(x) = sin(SQRT(x)), then the expectation exists, and it is equal to E(f) = 2 / Pi. (Prove it!)
The main result introduced at the beginning of this section, is rather intuitive but needs great care to prove it rigorously, including correctly stating the required assumptions on f and g to make it valid. Some cases might require working with non-Riemann integrals. Here we only provide the intuitive explanation.
Proof of the main result (sketch)
Here p, q and n are integers, with q greater than p. We are interested in the case where n tends to infinity. We approximate integrals using the Euler-Maclaurin summation formula. The approximations below become equalities as n tends to infinity.
We used the classic approximation of the harmonic series to make the logarithm terms appear. Note that for large values of k, g(k) is asymptotically equal to k. This was one of the requirements for the formula to be valid.
We also have:
Using the change of variable y = x / q in the first integral, and y = x / p in the second integral, we obtain:
Let us remark that:
q / g(qy) is asymptotically equivalent to 1 / y (for large values of y)
p / g(py) is asymptotically equivalent to 1 / y
both integrals diverge, so the impact of small values of y eventually vanishes in each integral separately
the difference between the two integrals converges
In view of this, we have:
This concludes the proof.
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