4.10
Antiderivatives
A function 'F' is called and
antiderivative of some function 'f' on interval 'I' if F(x) = f'(x) for
all ‘x’ in ‘I’.
Example) F(x) = 2x3
1. add 1 to 3
2. divide 2 by 4
f(x)
= 
x4 + C
(C = some constant)
Table of
Antidifferential Formulas
|
f(x) |
F(x) |
|
cf(x) |
cF(x) |
|
f(x) + g(x) |
F(x) + G(x) |
|
xn |
|
|
cosx |
sinx |
|
sinx |
-cosx |
|
sec2x |
tanx |
|
secxtanx |
sec2x |
|
|
|
Example) f’(x) = 8x3 + 12x + 3 f(1) = 6
f(x) = 
+ 
+ 3x + C
f(x) = 2x4 + 6x2
+ 3x + C
6 = 2(1)4 + 6(1)2 + 3(1) + C
6 = 2 + 6 + 3 + C
6 = 11 + C
C = -5
Example) f”(x) = x +
f’(1) =
2 f(1) = 1
f’(x) = 
+ 
f’(x) = 
x2 + 
+ C
f’(x) = 
x2 + 
+ 
2= 
(1)2 + 
(1)
+ C
f(x)
= 
+ 
+ 
+ C C= 
f(x) = 
+ 
x2.5 + 
x + C
f(x) =
+ 
x2.5 + 
x + -
Therom: If F is an antiderivative of f on
an interval I, then the most general antiderivative of f on I
is where C is an arbitrary constant
F(x) + C