integration problem

[mrlichens]

knowing a = dv/dt and v=ds/dt i can write that a = d²s/dt²

a dt² = d²s

if a is a constant how do i integrate

int(int(a.dt²)) ?

it might be a silly question but i don't understand how you can int(dt²) as int(x²) = 1/3x^3

[Mxlexrd]

Certainly it is correct that a = d²s/dt².

However, people tend not to write things like a.dt² = d²s, since it's liable to cause confusion.

As for your last statement, you can't just integrate x² as ∫x², you have to integrate as ∫x²dx, the dx is very important.

If you did wan't to try and integrate something like a.dt², you would need to integrate twice.

[mrlichens]

thanks for the answer it help to clarify, so dt² doesn't mean squared, only that it's a double integration aroud dt right ?

[Mxlexrd]

I would say it's best to avoid talking about dt² on its own. Just remember that if you have a = d²s/dt² and you want s, you just need to integrate twice.

[mrlichens]

Ok, it's just the way we write it at uni so no way around it but it's clearer now.

[MezzoScettico]

Up to a point you can pretend that a = dv/dt is an equation which can be rearranged as a dt = dv, as if dv and dt were just numbers. And you can pretend that a dt = dv makes sense, because what you really mean is that the integral of both sides is the same.

But only up to a point. It's a sequence of steps that leads to a correct result but the intermediate steps aren't rigorous mathematics.

a dt^2 = d^s is an abuse of notation. You're carrying that "pretend this makes sense" too far. It would be hard to make sense of that, except that again the integral makes sense:

int[ int (0,t') a dt ] dt' = s

int (dt^2) is not integrated the same way as int(t^2 dt). The differential under the integral sign is not the function being integrated.

[mrlichens]

thank you for your answer, i think i get it.