Antiderivatives
Let f(x) be continuous on [a,b]. If G(x) is continuous on
[a,b] and G¢(x) = f(x) for all x Î (a,b), then G is called an
antiderivative of f.
We can construct antiderivatives by integrating. The function
is an antiderivative for f since it can be shown that F(x)
constructed in this way is continuous on [a,b] and F¢(x) = f(x) for
all x Î (a,b).
Properties
Let F(x) be any antiderivative for f(x).
The process of finding antiderivatives is called
antidifferentiation or integration:
Properties of the Indefinite Integral
proof:
Let ò f(x) dx = F(x), where F(x) is an antiderivative
of f. Then
 (Linearity) ò[af(x)+bg(x)] dx = aò f(x) dx+bò g(x) dx.
proof:
We need only show that aò f(x) dx+bòg(x) dx is an antiderivative of ò [af(x)+bg(x)] dx:


d dx


é ë

a 
ó õ

f(x) dx+b 
ó õ

g(x) dx 
ù û




a 
d dx


é ë

ó õ

f(x) dx 
ù û

+b 
d dx


é ë

ó õ

g(x) dx 
ù û


 


 

Examples
 Every antiderivative of x^{2} has the form x^{3}/ 3 + C, since d/dx[x^{3}/ 3] = x^{2}.
 d/dx[ò x^{5} dx] = x^{5}.
Key Concept
If G(x) is continuous on [a,b] and G¢(x) = f(x) for all x Î (a,b), then G is called an antiderivative of f.
We can construct antiderivatives by integrating. The function
is an antiderivative for f. In fact, every antiderivative of f(x)
can be written in the form F(x)+C, for some C.

