Math 151 Section 5.7
Antiderivatives
An antideriva tive of f ( x ) is a function, F ( x ), for which F ' ( x )  f ( x ).
d
d
 F ( x )  C    F ( x )  for any constant, C , the most general
dx
dx
antideriva tive is F ( x )  C .
Since
Fill in the table :
f ( x)
antideriva tive
n  1
xn ,
1
x
ex
sin x
cos x
1
1  x2
1
1  x2
The linear property still holds for antiderivatives just as it does for
derivatives. That is, for any constants A , B constants and if
F ' ( x )  f ( x ),
G ' ( x )  g ( x ) then  AF ( x)  BG ( x) '  Af ( x)  Bg ( x) .
Examples:
Find the most general antiderivative.
1.
3x2  1
f ( x) 
x
2.
g ( x )  ( x  1) x
3.
f ( x )  5 x 1 3 
2
5
x
4.
Find the height of an object at t seconds if the object is
dropped from 200 ft. above ground. Use gravity  -32ft/s/s.
5 . an object is fired from the ground at 30  to the horizontal with an
initial velocity of 20 ft/s. If gravity  -32ft/s/s is the only accelerati on,
find the position v ector at t seconds after firing.