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Module 5.1:

Antiderivative - The Indefinite integral

Duy Tân University

Lecturer: Nguyen Thi Ngoc Bich

Natural Science Department

Chapter 5: Integration Module 5.1. Antiderivetive – The indefinite integral

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Chapter 5: Integration Module 5.1. Antiderivetive – The indefinite integral 2

5.1 Antiderivative - The Indefinite integral

1. Antiderivative

2. Rules for integrating

3. Practical applications

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Chapter 5: Integration Module 5.1. Antiderivetive – The indefinite integral

1. Antiderivative

- Antiderivative: A function F(x) for which for

every x in the domain of f is said to be an antiderivative of

f(x).

'( ) ( )F x f x

- Fundamental Property of Antiderivative: If F(x) is an antiderivative of the continuous function f(x), then any other antiderivative of f(x) has the form

G(x) = F(x) + C , for some constant C.

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Natural Science Department

Chapter 5: Integration Module 5.1. Antiderivetive – The indefinite integral

1. Antiderivative

We will represent the family of all antiderivatives of f(x) by using the symbolism:

which is called the indefinite integral of f

( ) ( ) ;f x dx F x C C const

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Chapter 5: Integration Module 5.1. Antiderivetive – The indefinite integral

2. Rules for integrating

for constantkdx kx C k 11

for all 11

x dx x C

1ln | | for all 0dx x C x

x 1

for all 0 kx kxe dx e C kk

( ) ( ) for constantkf x dx k f x dx C k [ ( ) ( )] ( ) ( )f x g x dx f x dx g x dx

+ Rules for integrating common functions:

+ Algebraic rules for indefinite integration:

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Chapter 5: Integration Module 5.1. Antiderivetive – The indefinite integral

3. Practical applications

Example 1: It is estimated that x months from now the polulation of a certain town will be changing at the rate of

people per month. The current population is 5000. What will be the population 9 months from now ? 4 6 x

Example 2: A manufacturer has found that marginal cost is dollars per unit when q units have produced.The total cost of producing the first 2 units is $ 900. What is the total cost of producing the first 5 units ?

23 5 100q q

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Chapter 5: Integration Module 5.1. Antiderivetive – The indefinite integral

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Natural Science Department