Lesson 01 Anti Differentiation

8/8/2019 Lesson 01 Anti Differentiation

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ANTIDERIVATIVES

OVERVIEW :

A comprehensive presentation of lessons relating to branch ofcalculus which is considered as the inverse of differential calculus, called integralcalculus. We were definitely amazed with the applications of differential calculussuch as the determination of the maximum and minimum values of a function,(both in curve tracing and stated problems), related rates problems, evaluation ofindeterminate forms of functions, and others. But another breakthrough in thestudents discovery of knowledge in mathematics is another chapter whichintegral calculus can offer: determination of area of irregular plane figures,volumes of solids of revolution (i.e., irregular solids), length of arc, etc.

Differential calculus starts with the basics of differentiation, so muchso with integration. This module provides the students with a clear view on basicintegration formulas on simple functions, and the generalized integrationformulas using simple algebraic substitution.

GENERAL OBJECTIVES:

After completing this module, the students are expected toaccomplish the following:

define antidifferentiation or integration;

discuss the different basic integration formulas; and

evaluate indefinite and definite integrals using basic integrationformulas and the generalized power formula.

HISTORICAL NOTES:

The Calculus is a phrase we use to denote that branch ofmathematics which deals with the properties of functions (curves) that areassociated with the limit process (continuity, differentiation, integration). Calculusis the introductory level of a more general branch of mathematics which is calledanalysis. Analysis deals generally with infinite processes and includes suchareas as real analysis, complex analysis, and differential equations. Kline callsthe calculus as one of the two greatest creations in the history of mathematics.

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LESSON 1: Indefinite Integral and Basic Integration

Formula

SPECIFIC OBJECTIVES:

At the end of this lesson, the student is expected to accomplish the following:

know the relationship between differentiation and integration;

identify and explain the different parts of the integral operation; and

perform basic integration by applying the power formula and theproperties of indefinite integrals.

DISCUSSION OF THE TOPICS:

At this point, students are already familiar with inverse operation in

Mathematics. To name a few, addition and subtraction, multiplication anddivision, raising to power and extracting roots, and some examples of inverseoperations.

In this lesson, we will discuss the inverse operation of differentiationwhich is called antidifferentiation or integration. Antidifferentiation or integrationis the process of finding the set of functions whose derivative or differential isgiven. The term integration is more often used than antidifferentiation but inthis module, it is not an issue whatever term is to be used.

Example 1.1Find a function whose derivative is x3 or its differential is x3 dx.

Solution:

Maybe by trial and error method, a reasonable guess to the unknown

function will be4x

4

1because if we get the derivative of

4x4

1, we will get x3 or

its differential, x3 dx. But what about4x

4

1+ 2?

4x4

1- 10?

4x4

1- 3 ?

2

Antiderivative (Integral)A function F is called an antiderivative or integral of the function f on

an interval I if F(x) = f(x) for every value of X in I.


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Actually any function in the form of4x

4

1+ C, where C is any

constant, is an antiderivative or integral of x3. With this reason, the antiderivativeor integral of a function always contain the C term.

Since integration or antidifferentiation is an operation, there must bean operator used. The operator for integration is the symbol (which looks likean elongated S that was derived from the first letter of the Latin word sum).

Thus, the statement a function whose differential is x3 dx is4x

4

1+C is

equivalent to Cx4

1dxx

43 += .

Properties of Indefinite Integral and Basic Integration Formula:

Some of the properties of indefinite integral and basic integrationformula, which need no proof from the fact that these properties are also knownproperties of differentiation are listed below.

Formula (iv) is the counterpart of the power formula in differentiationthat is why, it is also called as such in integration.

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Indefinite Integral

+= C)x(Fdx)x(f

if and only if F(x) = f(x)where f(x) is called the integrand;

C as the constant of integration;F(x) + C is the indefinite integral of f(x) dx; anddx indicates that x is the variable of integration.

Logically, the expression F(x) + C is called the indefinite integral sinceC has no definite value.

Properties of Indefinite Integral and Basic Integration Formula:

i) += Cxdx

ii) = dx)x(fkdx)x(kf ; k is a constantiii) +++=+++ ...dx)x(hdx)x(gdx)x(fdx...])x(h)x(g)x(f[

iv) ++=+

C1n

xdxx

1n

n ; n -1, Power Formula


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Example 1.2

Evaluate .dxx3

Solution:

Applying the power formula,

++=+

C13

xdxx

133

= C4

x4+

Example 1.3

Evaluate + dx)7x6x3(2

Solutions:

+=+ iii)(bydxxdxdxxdx)xx( 76376322

+= ii)(bydxxdxdxx 7632

Cx72

x6

3

x3 23++= (by ii and iv)

= x3 3x2 +7x + C

NOTE: Each of the terms, when evaluated, obtained a constant ofintegration C. But take note also that a constant added to another constant stillresults to a new constant. Thus, any one C appeared in the final answer.

Example 1.4

Evaluate dx)bax2(232

Solutions:

+=

+=

dxbdxxab4dxx4a

dx)bxab4xa4(dx)bax2(

62342

62342232

Cxb3xab4

5xa4 6

3352

++=

NOTE: a and b are constants. Different result is expected if dx is tobe replaced by da or db. Try to evaluate after replacing dx by da and then db.

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3. 3 2t

dt

4. dzzzz

+ 3

4/34

7

5. + dtat 3)5(

6. +

+dz

z

z

1

13

7. + dxxxe )2(

8. dxx

xxx

53

47

9. +++ dxxxx3 23

2754368

10. ++ dmmm 66 2

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Lesson 01 Anti Differentiation