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THIS DOCUMENT PROVIDES YOU WITH INFORMATION INCLUDING EXAMPLES YOU NEED TO KNOW ABOUT INTEGARL CALCULUS AT UNDEGRADUATE LEVEL
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6. INTEGRATIONINTRODUCTION
In mathematics, we are familiar with many pairs of inverse operations: addition and
subtraction, multiplication and division, raising to powers and extracting roots, taking
logarithms and finding antilogarithms, and so on. In this chapter, we discuss the inverse
operation of differentiation, which we call anti-differentiation or more commonly integration.
Integration is therefore the process of finding a function given its derivative or differential, i.e.
moving from f ' x to f (x) . For instance: a physicist who knows the velocity of a particle
might wish to know its position at a given time. An engineer who can measure the variable
rate at which water is leaking from a tank wants to know the amount leaked over a certain
period of time. A biologist who knows the rate at which a bacteria population is increasing
might want to deduce what the size of the population will be at some future time. In each
case, the problem is to find a function F whose derivative is a known function f . If such a
function F exists, it is called an antiderivative or an integral of f .
LEARNING OUTCOMES
On completion of this chapter, you will be able to:
Do anti-differentiation of basic algebraic functions and use the basic power and
quotient rules of integration with confidence.
Integrate all algebraic functions including rational functions.
Integrate transcendental functions.
Find the definite integral.
Apply integration to calculate areas and do word problems.
COMPILED BY T. PAEPAE
,6.1 STANDARD INTEGRATION
Why it is important to understand: Standard Integration
“Engineering is all about problem solving and many problems in engineering can be solved
using calculus. Physicists, chemists, engineers, and many other scientific and technical
specialists use calculus in their everyday work; it is a technique of fundamental importance.
Both integration and differentiation have numerous applications in engineering and science
and some typical examples include determining areas, mean and rms values, volumes of
solids of revolution, centroids, second moments of area, differential equations and Fourier
series. Besides the standard integrals covered in this chapter, there are a number of other
methods of integration covered in later courses (semester 2). For any further studies in
engineering, differential and integral calculus are unavoidable”. Bird, J., 2017. Higher
engineering mathematics. Routledge.
SPECIFIC OUTCOMES
On completion of this study unit, you will be able to:
Understand that integration is the reverse process of differentiation.
Perform anti-differentiation of basic algebraic functions and explain the meaning of
indefinite integrals.
Apply the basic power rule of integration with confidence and apply the basic quotient
rule of integration with confidence.
1
,6.1.1 The Process of Integration
In differentiation, if f ( x ) 2 x 2 then f ' x 4 x . Thus, the integral of 4 x is 2x 2 . By similar
reasoning, the integral of 2t is t 2 . In describing this reverse process, an elongated S
(called an integral sign), shown as is used to replace the words ‘the integral of’. Hence,
4x 2x 2t t
2 2
from above, and
dy
Remark, the differential coefficient indicates that a function of x is being differentiated
dx
with respect to x , the dx indicating that it is ‘with respect to x ’. In integration, the variable of
integration is shown by adding d (the var iable) after the function to be integrated. Thus:
4 x dx means ‘the integral of 4 x with respect to x ’, and
2t dt means ‘the integral of 2t with respect to t ’.
The function to be integrated (the coefficient of either dx or dt above) is known as the
integrand.
The Constant of Integration
As stated above, the differential coefficient of 2x 2 is 4 x , hence 4 x dx 2 x 2 . However, the
differential coefficient of 2 x 2 5 is also 4 x . Hence, 4 x dx is also equal to 2 x 2 5 . To
allow for the possible presence of a constant, whenever the process of integration is
performed, a constant ‘ C ’ is added to the result. Thus:
4x 2x C and 2t t 2 C
2
Where ‘ C ’ is called the constant of integration.
Note: Integration has one advantage that the result can always be checked by
differentiation. If the function obtained by integration is differentiated, we should get
back the original function (or the integrand).
2
, Example 6.1 Proving the Integrals
4 x dx x C
3 4
1. Prove that
Solution:
Differentiating the right side gives:
d 4
dx
x C 4x 3
Since 4x 3 is the coefficient of dx , we have proved that
3x dx x 3 C
2
2 3
2. Prove that x dx
3
x C
Solution:
Differentiating the right side gives:
d 2 2 23 2
3 1
x C x x
dx 3 3 2
This does equal the coefficient of dx , hence we have proved that
2 3
x dx
3
x C
6.1.2 Basic Rules of Integration
As in example 6.1, we can apply differentiation to prove the following:
1 dx x C
x2
x dx 2
C
x3
x dx 3 C
2
Inspection of the above equations shows that the answer has an exponent one greater than
that of the differential and the denominator equals this exponent. This can be generalised as:
x n 1
x dx C
n
Basic power rule:
n 1
3
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