(* Antiderivatives: #4, 38, 51, p. 273 *)

f[x_]:=8x^9-3x^6+12x^3
Integrate[f[x],x]

fpp[x_]:=20x^3+12x^2+4
Integrate[fpp[x],x]
fp[x_]:=4 x+4 (x^3)+5 (x^4)+c
Integrate[fp[x],x]
f[x_]:=c x+2 (x^2)+(x^4)+(x^5)+8
f[x_]:=-7x+2 (x^2)+(x^4)+(x^5)+8
f[0]
f[1]
f''[x]

v[t_]:=Sin[t]-Cos[t]
Integrate[v[t],t]
s[t_]:=-Cos[t]-Sin[t]+1
s[0]