Antiderivatives

Antiderivatives

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.

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.

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

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

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 ∫.

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.

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

Basic Integration Rules

Basic Integration Rules cont’d

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

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.

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

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