ANTIDERIVATIVES
Definition
Suppose that F'(x) =
f(x). Then F(x)
is called an antiderivative
of f(x).
Examples

An antidervative of sin(x) is
cos(x).

An antiderivative of x^{2}
is 1/3 x^{3
}
Differential Equations
Suppose we want to find the solution to
dy/dx = f(x)
This is another way of asking for an antiderivative of f.
Example
Find the solution to the differential equation
dy/dx = 3x^{2}
 4x + 2
Solution
We find the antiderivative of
3x^{2}
 4x + 2
We can find this antiderivative by finding the antiderivative of x^{2},
x, and 2 individually
and then subtracting and adding. We have
Function 
x^{2} 
x 
2 
Antiderivative 
1
x^{3}
3 
1
x^{2}
2 
2x 
= x^{3}
 2x^{2} + 2x
Notice that since the derivative of a constant is zero, adding a constant of
an antiderivative results in another antiderivative for the same
function. We can write the final answer as
x^{3}
 2x^{2} + 2x + C
where C represents any constant.
Notation
The antiderivative of f(x) will be denoted by
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