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The problem related to my question is that it should be impossible to integrate such a function or I am just not familiar with doing it for that example.

Maybe it helps to understand my question, I figgured out a Disevenly accelerated movement and know I have to depict the in a diagram and also the velocity and the way.

I am sure that the diagram actually is not depicting an acceleration. But the graph is similar, which is crucial for that case.

enter image description here

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  • $\begingroup$ [...] it should be impossible to integrate such a function [...] You can integrate such a function piecewise. For the shown diagram, you might do something like $\int_0^{57.5} \dots \mathrm{d}x = \int_0^{10} \dots \mathrm{d}x + \int_{10}^{20} \dots \mathrm{d}x +\int_{20}^{35} \dots \mathrm{d}x + \int_{35}^{57.5} \dots \mathrm{d}x$. $\endgroup$ – scaphys Feb 20 '19 at 21:19
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I'm not exactly sure about what you're asking, but I'll try to guide you through an explanation.

Position, velocity and acceleration are all functions of time. You can express it as $x(t)$, $v(t)$ and $a(t)$. You might also try to think about the relations between them. Velocity is the change in position over a time interval, and acceleration is the same but with velocity. If you know some calculus you could say:

$v=dx/dt\quad$ and $\quad a=dv/dt$

So if you know the acceleration of an object $a(t)$, you can find the velocity by integrating with respect to time:

$$v(t) = \int_{t_0}^t a(t) dt $$

And by doing so you get a function of time you can plot as in the image you privided.

Same with position:

$$x(t)= \int_{t_0}^t v(t) dt$$

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