4.1a Antiderivatives and Indefinite Integration Write the general solution of a differential equation Use indefinite integration for antiderivatives Use basic integration rules to find antiderivatives Find a particular solution of a differential equation
Suppose you were asked to find the function F(x) whose derivative is f(x) = 3x 2 . What would you come up with? How about if the derivative was f(x) = x 2 ? How about if the derivative was f(x) = 2x 4 ?
Notice that F(x) is called an antiderivative, not the antiderivative. See why: F 1 (x) = x 3 F 2 (x) = x 3 + 2 F 3 (x) = x 3 – 5 Each would have the same derivative, f(x) = 3x 2 In other words, any constant added on would give same results.
You get a whole FAMILY of antiderivatives by adding a constant C to the known antiderivative. A point on the curve of the antiderivative might be needed to nail down what the constant is for a specific case. C is called the constant of integration . Knowing that D x [x 2 ] = 2x, all antiderivatives of f(x) = 2x would be represented by G(x) = x 2 + C ; this would be called the general antiderivative , and G(x) is the general solution of the differential equation G’(x) = 2x A differential equation in x and/or y is an equation that involves x, y, and the derivatives of y. y‘ = 3x and y’ = x 2 +1 are examples of differential equations.
Ex 1 p. 249 Solving a differential equation Find the general solution to the differential equation y’ = 3 Solution: You need to find a function whose derivative is 3. How about y = 3x? With info from theorem 4.1, the general solution could be y = 3x + C.
Notation for antiderivatives: When solving a differential equation of the form It is easier to write in the equivalent dy = f(x)dx form. The operation of finding all solutions of differential equations is called antidifferentiation or indefinite integration . Notation: (read antiderivative of f with respect to x. Notice that dx tells you what variable you are integrating with respect to.
Integration is the “inverse” of differentiation Differentiation is the “inverse”of integration