ch-8, integrations or anti-derivatives 2 [ pdf rough file ]

7/28/2019 Ch-8, Integrations or Anti-Derivatives 2 [ PDF Rough File ]

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BIO-MEDICAL ENGINEERING GUIDE.INC

2013

[HIGH-SCHOOL-

MATHEMATICS,

INTEGRATION,CH-8][INTEGRATIONS OR ANTI-DERIVATIVES OR

INDEFINITE-INTEGRAL]

BY MOHAMMAD-SIKANDAR-KHAN-LODHI

W W W . M E D I C A L - I M A G E - P R O C E S S I N G . B L O G S P O T . C A B Y S I K A N D E R - L O D H I


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[HIGH-SCHOOL-MATHEMATICS, INTEGRATION,CH-8]

W W W . M E D I C A L - I M A G E - P R O C E S S I N G . B L O G S P O T . C A B Y

S I K A N D E R - L O D H I

Page 2-OF-

ANTI-DERIVATIVE=INDEFINITE-INTEGRAL=INTEGRATION:-

8.1 ANTI-DERIVATION OR INTEGRATION:-

In the previous chapter, we discussed the problem of finding the derivativeof a given function

However, many of the most important applications of Calculus lead to theinverse problem of finding the function when its derivative is given .

The required function is called an anti-derivative or an indefinite integral ofthe given derivative.

The process of finding it is called anti-derivation or integration. To be precise, let f,unctions from A to . then FF is called an

antiderivative of f if F(x) exist and * ]fir all x A. Often, we shall say F(x) is an anti-derivative or an integral of f(x) over A to

mean that F:A is an antiderivative of f:A. Originally,the concept of antiderivative or integral of a function f, was

developed in order to find a certain area.

C=constant of Integrations=Arbitrary-constant, and f(x)=integrand.[

];

The expression

[ ];Is called the indefinite integral, or just the integral of f(x) with respect to x, and

is denoted by

[

];

Henceforth,wherever,we write it will be assumed that anantiderivative of f(x) exists.

From now on we shall always use C for the constant of integration. We mention,without proof , the following important result to be used in

the evaluation of integrals.


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Page 3-OF-

a) [ ];b) [ ];

Provided that the antiderivatives of the function f and g exist.8.1.1 Formulas For Integrals Of The Form:-

[ ];Where the integrands have properly defined domains.

i. As [ ];we have [ ]; provided[ ];

ii. Again , as [

];we

have [ ]; provided [ ];iii. Finally, as [ ]; we have

[ ];provided [ ];Example:

1. [ ];2. [ ];

8.1.2 Definite-Integral:-

If a & b are two real numbers such that ab,we defined the quantity.[ ];As the definite-integral of f(x) between a and b, and write it as

[ ];Provided that [a,b] is contained in the domain of f.thus if

[ ];Then


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Page 4-OF-

[ ];[ ];[

];

[ ];The real numbers a & b are called the Lower & Upper limits of the

definite-integral.

The Real-number represents the area A bounded by the curve y=f(x) and

the lines x=a, x=b, see figure 8.1 below .

The shaded portion shown in figure is the area A .

[ ];Fig 8.1

Standard Formulae For Integration :-

In the chapter on differentiation we get a large number of standardformulas for differentiation.


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Page 5-OF-

Using these , we now get the following set of standard formulae forintegration.

S.No f(x) 1 2 3 4 5

6 7 8 9 10 11 12 13

14 15 16 17

18 19

20 21 22


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Page 6-OF-

23 24

25 26 [ ];8.2 INTEGRATION BY SUBSTITUTION:-

Its sometimes found that a change in variable x in * ] bymeans of a substitution of the form [ ]( ) brings the integralto a standard form , or to a form easier to deal with .

This is particularly so if the given integral is of the form [ ], for then from properties of differentials,we know that [

], so that we get

[ ], Obviously,the integral on the right is much easier to calculate thanthat on the left.we shall illustrate this method by some examples.

8.2.1 INTEGRALS OF cos(ax+b), AND sin(ax+b), (a,b BEING CONSTANTS AND a0

).

1. [ ];[ ];Let, [ ];We differentiating w.r.t x

[ ];


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Page 7-OF-

[ ];So our new integral becomes

[

];

[ ];[ ];[ ];

2. [ ]:-----------

Ex # 1:Find

[ ];Solution:

Let: [ ];Differentiating w.r.t x

[ ];[

];

So, integral become

[ ];

[ ];[

];[ ];--------------finished here-

Ex# 2

Find [ ];Solution:


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Page 8-OF-

[ ];Let :

[ ];Differentiating w.r.t theta

[ ];[ ];

Our given integral become

[

];

[ ];[ ];[ ];[ ];[ ]; Answer.---------------finished-heree--------

Example # 4 :-

Find [ ];Solution:

[ ];{:. }{:.

}

Let , [ ];[ ];[ ];[ ];


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Page 9-OF-

[ ];[ ];[ ]; Answer.------------finished-here-------

Ex # 5 :

Find [ ];Solution:

[ ];We do like that

[ ];[ ];Let:

[ ];[ ];[ ];[ ];[

];

[ ];[ ];[ ];[ ];[ ];[ ];[ ];[

];

[

];

[

];

[ ];[ ];[ ];[ ];[ ];[ ];[ ]; Answer.-------finished-here--------------

8.2.2 Substitution in Definite-Integrals :- When a substitution is made in a definite-integral, the limits of integration should also be changed. The new limits are obtained as the values of new variable when the old variable equals the old limits. We illustrate this by some examples. As follow :

Ex # 1 :- Evaluate: [ ];Solution:-

We have a given definite integral in below eq-A.


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[ ];Let ,

[ ];[

];

[ ];[ ];For new Lower limit:- when old Lower limit is [ ][ ];[ ]; its our new lower limit .For New Upper-limit:- When old Upper-limit is [ ]:-[

];

[ ]; its our new Upper limit .So, our given integral become.

[ ];[ ];[

];

[

];

[

];[ ];[ ]; Answer.-----------Finished-here-----------

Example # 2 :-

Find [

];Solution:-

We have a given definite integrals

[ ];Let, [ ];


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Page 11-OF-

Differentiating on each s ide with respect to *w.r.t+ x

[ ];[ ];For new Lower-Limit , when old Lower-limit is [

]:-

[ ]; [ ];[ ];For new Upper-Limit , when old Upper-limit is [ ]:-[ ];[ ];

[

];

[ ];

[

];

[

];

[

]; Answer.------------------Finished-here-----------

8.2.3 TRIGONOMETRIC-SUBSTITUTIONS:

If the integrand contains expressions of the form in below table then we use the following Trigonometric substitutions:

S.No Integrand Trigonometric-substitutions:

1 2 3 -------------------

Ex # 3 : Evaluate the following :

[ ];Solution:

[ ];Let: [ ];Differentiating


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[ ];For Radical [ ]:-[ ];[

];

[ ];[ ];[ ];[ ];[ ]; its our Radical .The limit for are:For New Lower-Limit: [ when Old Lower limit is ( ) ][ ]; [ ];[ ];

For New Upper-Limit: [ when Old Upper-limit is ( ) ][ ]; [ ];[ ];

Henced our integral become.

[ ];[ ];[ ];[ ];[ ];[

];

[ ];[ ];[ ];answer----------finished-here------


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8.3 INTEGRATION BY COMPLETING THE SQUARES:-

If the integrand involves a quadratic expression in the denominator or under a radical,a useful device is to complete the square.Ex # 1: Find [ ];Solution:

[ ];[ ];[ ];[ ];[ ];[

];

[ ];[ ];Our above equation becomes.

Let:

[ ];[

];

[

];

[

];

[

];

[

];

[ ];

[

];

[

]; Answer.


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-----------finished-here----------

Ex # 2 :- Find [ ];Solution:

[

];

------------Start from p.g no 235 from text book .--- ---

ch-8, integrations or anti-derivatives 2 [ pdf rough file ]

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