v[t_]:=Sin[Om*t]*Cos[Om*t]^2
Om=2
Plot[v[t],{t,0,2Pi}]

Plot[Integrate[v[t],{t,0,x}],{x,0,2Pi}]

Integrate[Sin[x]^2*Cos[x]^3,x]

Integrate[Sin[Sqrt[x]]^3/Sqrt[x],x]

Integrate[Cos[x]^2,{x,0,Pi/2}]

Integrate[Cos[Pi*x]*Cos[4*Pi*x],x]