6 Integration Antiderivatives and the Rules of Integration Integration by Substitution Evaluating Definite Integrals

(6.1): Antiderivative A function F is an antiderivative of f on an interval I if for all x in I. Ex. is an antiderivative of Since

Theorem Let G be an antiderivative of f. Every antiderivative F of f must be of the form F(x) = G(x) + C, where C is a constant. Notice are all antiderivatives of

read “the integral of f,” The process of finding all antiderivatives of a function is called integration. The Notation: read “the integral of f,” means to find all the antiderivatives of f. Indefinite integral Integral sign Integrand

Basic Rules Rule Example

Basic Rules Rule Ex. Find the indefinite integral

Initial Value Problem To find a function F that satisfies the differential equation and one or more initial conditions. Ex. Find a function f if it is known that: Gives C = 3

(6.2): Integration by Substitution Method of integration related to chain rule differentiation. Ex. Consider the integral: Sub to get Integrate Back Substitute

Integration by Substitution Steps: 1. Pick u = f (x), often the “inside function.” 2. Compute 3. Substitute to express the integral in terms of u. 4. Integrate the resulting integral. 5. Substitute to get the answer in terms of x.

Ex. Find Pick u, compute du Sub in Integrate Sub in

Ex. Find

Ex. Find

(6.4): Fundamental Theorem of Calculus Let f be continuous on [a, b]. Then where F is any antiderivative of f.

Evaluating the Definite Integral Ex. Evaluate