∫a∫a dt=vdt=v

∫v∫v dt=sdt=s

where aa = acceleration, vv = velocity, and ss = displacement

Beginning with the first integral:

∫a∫a dtdt

=at+c=at+c

=v=v

v=c+atv=c+at

Here, cc represents some arbitrary constant. When it comes to motion, cc would represent the initial velocity of the moving body. Commonly, that is represented as uu, so I’m going to change it to

v=u+atv=u+at

Now the second integral:

∫v∫v dtdt

=∫u+at=∫u+at dtdt

=ut+12at2=ut+12at2

=s=s

So,

s=ut+12at2s=ut+12at2

These are some of the equations of motion that can be derived using integration. Of course, there are other equations of motions that I haven’t mentioned, like

v2=u2+2asv2=u2+2as

and

s=(v+u)t2s=(v+u)t2