Lecture7| 1

Lecture7| 1

Chapter 3 Integration

In this chapter we explore the following situation

This is more-or-less the same as

Lecture7| 2

EXAMPLE. Consider an experiment.

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

feet. Assume the acceleration is feet per second².

Acceleration-velocity relation :

Initial velocity :

Thus, we need to find a velocity that has

and .

Lecture7| 3

Once having found , the height can be found from the velocity-position relation

and the initial height .

Lecture7| 4 3.1. Antiderivatives and Indefinite Integrals

Consider the function . We want to find a function satisfying the equation

Since , we get one function

is called an antiderivative of .

Definition. For a function , if satisfies

then it is called an antiderivative of .

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Question Find an antiderivative for 1.

2.

3.

4.

5.

Lecture7| 6 For , observe that addition to

one can show that

are also antiderivatives of . In fact,

Theorem. If is an antiderivative of , then any antiderivative of has the form

where is any number.

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 gives all antiderivatives.

 It is called the antiderivative of , denoted by

 It is also called the indefinite integral of .

 The process of finding is called integration

Lecture7| 8 EXAMPLE. Evaluate

1.

2.

3.

4.

5.

Lecture7| 9 Basic Integration Rules

Note The is applied only once at the end:

Lecture7| 10 Question Evaluate the antiderivatives

Lecture7| 11 EXAMPLE. Evaluate the following indefinite

integrals.

Lecture7| 12 EXAMPLE. Find the antiderivative of

Lecture7| 13 EXAMPLE. Solve the differential equation

for a function satisfying .

Lecture7| 14 Basic Integration Rules (Trigonometric)

Lecture7| 15 Question Evaluate the antiderivatives

Lecture7| 16 EXAMPLE. Evaluate the indefinite integrals

Lecture7| 17 Basic Integration Rules (Inverse Trigonometric)

Lecture7| 18 EXAMPLE. Evaluate