AntiderivativesAntiderivativesIndefinite IntegralsIndefinite Integrals

DefinitionDefinition

A function A function FF is an is an antiderivativeantiderivative of of ff on an on an interval interval II if if F’(x) = f(x)F’(x) = f(x) for all for all xx in in II..

Example:Example:

F(x) = xF(x) = x33 when f(x) = 3x when f(x) = 3x22

In other words, we are going backwards – given In other words, we are going backwards – given the derivative, what was the original function?the derivative, what was the original function?

Representation of AntiderivativesRepresentation of Antiderivatives

If If FF is an antiderivative of is an antiderivative of ff on an interval on an interval II, , then then GG is an antiderivative of is an antiderivative of ff iff iff

G(x) = F(x) + C, for all G(x) = F(x) + C, for all xx in in II

where C is a constantwhere C is a constant

Representation of AntiderivativesRepresentation of Antiderivatives

The constant The constant CC is called the constant of is called the constant of integration. integration.

The family of functions represented by The family of functions represented by GG is the general antiderivative of f and the is the general antiderivative of f and the answer is the general solution of the answer is the general solution of the differential equation.differential equation.

Differential EquationDifferential Equation

A differential equation in A differential equation in xx and and yy is an is an equation that involves equation that involves x, y, x, y, and derivatives and derivatives of of yy..

Find the general solution of the differential Find the general solution of the differential equation y’ = 2.equation y’ = 2.

Differential EquationsDifferential Equations

Find the general solution of the differential Find the general solution of the differential equation y’ = xequation y’ = x22

Find the general solution of the differential Find the general solution of the differential equation equation

2

1'yx

Notation for AntiderivativesNotation for Antiderivatives

When solving a differential equation of the When solving a differential equation of the formform

The operation of finding all solutions of this The operation of finding all solutions of this equation is called antidifferentiation (or equation is called antidifferentiation (or indefinite integration)indefinite integration)

( ) ( )dy

f x or dy f x dxdx

Notation for AntiderivativesNotation for Antiderivatives

The general solution is denoted byThe general solution is denoted by

Where Where f(x) f(x) is the integrand, is the integrand, dxdx is the variable is the variable of integration and of integration and CC is the constant of is the constant of integration.integration.

This is an antiderivative of f with respect to x.This is an antiderivative of f with respect to x.

( ) ( ) .y f x dx F x C

Basic Integration RulesBasic Integration Rules

Differentiation and antidifferentiation are Differentiation and antidifferentiation are inverse functions (one “undoes” the other)inverse functions (one “undoes” the other)

The basic integration rules come from the The basic integration rules come from the basic derivative rulesbasic derivative rules

Basic Integration RulesBasic Integration Rules

Rules:

Applying the Basic Integration Applying the Basic Integration RulesRules

Evaluate the indefinite integralEvaluate the indefinite integral

3x dx

1

2

2

3 Constant multiple rule

3 Rewrite

3 Power Rule 12

3Simplify

2

x dx

x dx

xC n

x C

Rewriting Before IntegratingRewriting Before Integrating

Original Integrand REwrite Integrate Simplify

23

3 2

1 1

2 2

xdx x dx C C

x x

3

1 322 22

3 32

xx dx x dx C x C

2sin 2 sin 2 cos 2cosx dx x dx x C x C

Integrating Polynomial Integrating Polynomial FunctionsFunctions

1dx dx x C 3( 2)x dx4

24

xx C

Integrating PolynomialsIntegrating Polynomials

2 2 3x x dx 3

2 33

xx x C

Rewriting Before IntegratingRewriting Before Integrating

We have no division rules at this pointWe have no division rules at this point When dividing by a monomial, put the When dividing by a monomial, put the

monomial under each of the monomials in monomial under each of the monomials in the numeratorthe numerator

2 1x xdx

x

Rewriting Before IntegratingRewriting Before Integrating

2 1x xdx dx dxx x x

3 1 1

2 2 2x dx x dx x dx

5 3 1

2 2 2

5 3 12 2 2

x x xC

Rewriting Before IntegratingRewriting Before Integrating

5 3 1

2 2 22 22

5 3x x x C

Rewriting Before IntegratingRewriting Before Integrating

2

sin

cos

xdxx

1 sin

cos cos

xdx

x x

sec tanx x dxsec x C

Initial Conditions and Particular Initial Conditions and Particular SolutionsSolutions

A particular solution allows you to A particular solution allows you to determine the C in a differential equation. determine the C in a differential equation. In order to do this, initial conditions must In order to do this, initial conditions must be given.be given.

Finding a Particular SolutionFinding a Particular Solution

23 1dy

xdx

23 1dy x dx

23 1dy x dx 3 1y x x C General solution

Finding a Particular SolutionFinding a Particular Solution

(2) 4Given F

3(2) 2 2 8 2 4 2Find C F C C C

3. . . ( ) 2So F x x x

Solving a Vertical Motion Solving a Vertical Motion ProblemProblem

A ball is thrown upward with an initial A ball is thrown upward with an initial velocity of 64 feet per second from an velocity of 64 feet per second from an initial height of 80 feetinitial height of 80 feet

a. Find the position function giving the a. Find the position function giving the height height ss as a function of the time as a function of the time tt..

b. When does the ball hit the ground?b. When does the ball hit the ground?

On-line HelpOn-line Help

Tutorial

Video Help