ANTIDERIVATIVES AND INDEFINITE INTEGRATION

Section 4.1Section 4.1

When you are done with your homework, you should

be able to…– Write the general solution of a

differential equation– Use indefinite integral notation for

antiderivatives– Use basic integration rules to find

antiderivatives– Find a particular solution of a

differential equation

Thales lived in 600 BC. He is famous for being the first

person to…

A. …use deduction in mathematics.

B. …measure the size of the earth.C. …characterize the conic sections.D. All of the above.

ANTIDERIVATIVES

• A function F is an antiderivative of f on an interval I if for all x in I.– Why does the definition use “an

antiderivative” instead of “the antiderivative”?

F x f x

Theorem: Representation of Antiderivatives

• If F is an antiderivative of f on an interval I, then G is an antiderivative of f on the interval I if and only if G is of the form , for all x in I where C is a constant.– How is this theorem different from the

last definition?

G x F x C

Some terms to be familiar with…

• The constant C is called the constant of integration.

•  The family of functions represented by G is the general antiderivative of f.

• is the general solution of the differential equation G x F x C

.G x F x

DIFFERENTIAL EQUATION

• A differential equation in x and y is an equation that involves x, y and derivatives of y.– Examples:

and 5y x 4 2y x

Solving a Differential Equation

• Find the general solution of the differential equation .– Solution: We need to find a function

whose derivative is 6. The function has a derivative of 6. Using the previous theorem, we write the general solution as .

6y

6y x

6y x C

Solve the differential equation

A. B. C. D. Both A and C

5y

5y x C

5y C 5y x

Solve the differential equation

A. B. C.

D.

2y x

33y x C

2y x C

3y x C

3

3xy C

NOTATION FOR ANTIDERIVATIVES

• When solving a differential equation of the form , we solve for , giving us the equivalent differential form .– This means you isolate dy by

multiplying both sides of the equation by dx. It is easier to see if you write the left side as instead of

dy f xdx

dy f x dx

dy

dydx

.y

Notation continued…• The operation of finding all

solutions of this equation is called antidifferentiation or indefinite integration and is denoted by an integral sign . The general solution is denoted by

Variable ofIntegration

Constant ofIntegratInt ionegrand

y f x dx F x C

Solve the differential equation

A. B. C.

D.

2secy x

2tany x C

secy x C tany x C

sec tany x x C

SOLVING A VERTICAL MOTION PROBLEM

The Grand Canyon is 1800 meters deep at its deepest point. A rock is dropped from the rim above this point. Write the height of the rock as a function of the time t in seconds. How long will it take for the rock to reach the canyon floor?

Vertical motion continued…• Use as the

acceleration due to gravity. Neglect air resistance. Recall that represents initial velocity, represents initial position. So

. How did we get from the acceleration function to the position function?

29.8 /a t m s

0v0s

20 04.8f t t v t s

Continued…

initial velocity is zero

2

canyon flooras position zero

0 4.8 0 1800t

2

2

4.8 180018004.8

9.2

t

t

t s