What is a "product integral" and how would I evaluate one?

Question

So I was watching some math videos for fun when I came across this video by Maths$505$. Now, on the thumbnail, there was an oddly structured integral^[$1$] and differential equation. (Note that my question is about the integral, the oddly structured differential equation makes sense to me) Now, about the integral, it seems to be structured in the form$$\int f(x)^{dx}$$which is weird, since I don't think that usually, integrals could be structured in this form. Now, watching the video, it seems that that type of integral is called a "product integral", but I'm not sure how I would be able to evaluate one in this specific form.

My question

How would I evaluate an integral of the form $\displaystyle\int f(x)^{dx}$? I'm not sure how this type of integral works, and I wasn't able to find any info from searching on Google.

Notes

^[$1$]The integral in the thumbnail in question is$$\int_0^{+\pi/2}(\sin(x))^{dx}$$and here is a screenshot of the thumbnail if anyone wants to see (and if the image doesn't load for you)

geometric integrals they are just $\exp(\int_a^b\log f(x),\mathrm{d}x)$. — user10354138, Oct 09 '23 at 14:15
@user10354138 ... for scalar-valued function $f$, the product integral is, indeed, $\exp(\int_a^b\log f(x),\mathrm{d}x)$. So it is not of much interest. However we do see product integrals for matrix-valued functions $f$, and if the values $f(x)$ do not commute, we cannot do it as $\exp(\int_a^b\log f(x),\mathrm{d}x)$. See https://www.amazon.com/Integration-Application-Differential-Encyclopedia-Applications/dp/0521302307 — GEdgar, Oct 09 '23 at 14:34
Why can’t we just have the logarithm of the matrix valued function $f$? — Тyma Gaidash, Oct 09 '23 at 15:06
See this answer of mine for a brief contextualization : https://math.stackexchange.com/questions/4605603/does-the-formula-lim-delta-x-rightarrow-0-sqrt-delta-x-fracfx-delta/4605755#4605755 — Abezhiko, Oct 12 '23 at 10:13

What is a "product integral" and how would I evaluate one?

My question

Notes

0 Answers0