Section 3.9: anti-derivatives

Calculus I
Andy Long

#3, p. 273

f[x_]:=1/2+3/4x^2-4/5x^3
Integrate[f[x],x] (* Notice that Mathematica doesn't add the arbitrary constant C: *)

#20, p. 273

f[x_]:=x + 2 Sin[x]
F[x_]=Integrate[f[x],x] (* Notice that Mathematica doesn't add the arbitrary constant C: 
find the constant that satisfies 
f[0]+constant=6 
*)
constant=6-F[0]
F[x]+constant

#33, p. 273

fpp[x_]:=-2+12x-12x^2
fp[x_]=Integrate[fpp[x],x] (* Notice that Mathematica doesn't add the arbitrary constant C: 
find the constant that satisfies 
fp[0]+constant=12 
*)
constant=12-fp[0]
f[x_]=Integrate[fp[x]+constant,x] (* 
f[0]+constant=4
*)
constant=4-f[0]
f[x]+constant

#41, p. 273

fp[x_]:=2x+1
f[x_]=Integrate[fp[x],x] (* Notice that Mathematica doesn't add the arbitrary constant C: 
find the constant that satisfies 
f[1]+constant=6 
*)
constant=6-f[1]
f[x]+constant

#70, p. 275

g=32 (* feet per second per second*)
fpp[t_]:=If[t<3,60 t,
If[t<17,-g,
If[t<22,(-18-fp[17])/5,0]]]
fp[t_]=Integrate[fpp[t],t] 

constant=0-fp[0]
f[t_]=Integrate[fp[t]+constant,t] 
constant=0-f[0]
f[t]+constant
Plot[f[t],{t,0,46}]