I was watching an introductory video series on differential equations by Gilbert Strang. In one of the first videos there was a lesson on calculus you need to know to understand the rest of the series. After watching, I had a quick question about notation for integration. Specifically, The fundamental theorem of calculus was used to shown that
$$y(t) = \int_0^t e^{t-s} q(s) ds$$ solves $$\frac{dy}{dt}=y+q(t)$$
Using the product rule. The thing that confuses me is what (s) refers to in this context. I'm comfortable with basic integration from school, integrating e.g. x^2 etc. In the above example, I don't understand how the integral is between 0 and t, yet there is also an (s) term in there. Similarly, I don't understand how the integral has a ds at the end, shouldn't the whole thing not be done with respect to t?
Hopefully my question makes sense. video and timestamp for context https://www.youtube.com/watch?v=f0BxAtprWts&t=240s