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
These will need to be memorized now too!
Ex 2 Applying the basic integration rules Find the antiderivatives of 5x Solution: Since C represents ANY constant, we could write in the simpler form
Ex. 3 p. 251 Rewriting before Integrating
Ex 4 p 252 Integrating Polynomial Functions
Ex 5 p. 252 Rewriting before Integrating
Ex 6 p252 Rewrite before Integrating
One of the most important steps to integration is REWRITING the integrand in a form that fits the basic integration rules. 4.1a p. 255/ 3-33 mult of 3, 35-45 odd