Section 17.4 Integration LAST ONE!!! Yah Buddy!

 A physicist who knows the velocity of a particle might wish to know its position at a given time.  A biologist who knows the rate at which a bacteria population is increasing might want to deduce what the size of the population will be at some future time. Introduction

 In each case, the problem is to find a function F whose derivative is a known function f.  If such a function F exists, it is called an antiderivative of f. Antiderivatives Definition A function F is called an antiderivative of f on an interval I if F’(x) = f (x) for all x in I.

Find the integral. (Find the antiderivative.) = ?

 If F is an antiderivative of f on an interval I, then the most general antiderivative of f on I is F(x) + C where C is an arbitrary constant. Theorem Antiderivatives  Going back to the function f (x) = x 2, we see that the general antiderivative of f is ⅓ x 3 + C.

Notation for Antiderivatives  The symbol is traditionally used to represent the most general an antiderivative of f on an open interval and is called the indefinite integral of f.  Thus, means F’(x) = f (x) because the derivative of is

Every antiderivative F of f must be of the form F(x) = G(x) + C, where C is a constant. Example: Represents every possible antiderivative of 6x. Constant of Integration

Example: Power Rule for the Indefinite Integral

Indefinite Integral of e x and b x Power Rule for the Indefinite Integral

Sum and Difference Rules Example:

Constant Multiple Rule Example:

Integration by Substitution Method of integration related to chain rule. If u is a function of x, then we can use the formula

Example: Consider the integral: Sub to getIntegrateBack Substitute Integration by Substitution

Example: Evaluate Pick u, compute du Sub in Integrate

Example: Evaluate

Examples on your own:

Find the integral of each: 1.)2.) 3.)4.)

Find the integral of each: 5.)6.) 7.)8.)

Find the integral of each: 9.)10.) 11.)12.)

Find the integral of each: 13.)14.)