1. Antiderivatives

2. Antiderivatives

3. Antiderivatives
Note that F is called an antiderivative of f, rather than the antiderivative of f. To see why, observe that
are all derivatives of f (x) = 3x2. In fact, for any constant C, the function F(x)= x3 + C is an antiderivative of f.

4. Antiderivatives
The family of functions represented by G is the general antiderivative of f, and G(x) = x2 + C is the general solution of the differential equation
G'(x) = 2x Differential equation
A differential equation in x and y is an equation that involves x, y, and derivatives of y.
For instance, y' = 3x and y' = x2 + 1 are examples of differential equations.

5. Example 1 – Solving a Differential Equation
Find the general solution of the differential equation y' = 2.

6. Example 1 – Solution
cont’d
The graphs of several functions of the form y = 2x + C
are shown
Figure 4.1

7. Notation for Antiderivatives
When solving a differential equation of the form
it is convenient to write it in the equivalent differential form
The operation of finding all solutions of this equation is
called antidifferentiation (or indefinite integration) and is
denoted by an integral sign ∫.

8. Notation for Antiderivatives
The general solution is denoted by
The expression ∫f(x)dx is read as the antiderivative of f with respect to x. So, the differential dx serves to identify x as the variable of integration. The term indefinite integral is a synonym for antiderivative.

9. Basic Integration Rules
The inverse nature of integration and differentiation can be verified by substituting F'(x) for f(x) in the indefinite integration definition to obtain
Moreover, if ∫f(x)dx = F(x) + C, then

10. Basic Integration Rules

11. Basic Integration Rules
cont’d

12. Basic Integration Rules
The general pattern of integration is similar to that of differentiation.

13. Initial Conditions and Particular Solutions
You have already seen that the equation y = ∫f(x)dx has
many solutions (each differing from the others by a
constant).
This means that the graphs of any two antiderivatives of f
are vertical translations of each other.

14. Initial Conditions and Particular Solutions
For example, Figure 4.2 shows the
graphs of several antiderivatives
of the form
for various integer values of C.
Each of these antiderivatives is a solution
of the differential equation
Figure 4.2

15. Example 8 – Finding a Particular Solution
Find the general solution of
and find the particular solution that satisfies the initial
condition F(1) = 0.