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MT1001 Introductory MathematicsIntegration Lecture Notes 1
Tom Coleman
November 7, 2018
1 This work is licensed using a CC BY-NC-SA 4.0 license.
,Contents
1 Definite integrals 3
1.1 What is integration? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
1.2 Definite integrals as areas . . . . . . . . . . . . . . . . . . . . . . . . . . 4
1.3 Properties of definite integrals . . . . . . . . . . . . . . . . . . . . . . . . 9
2 The Fundamental Theorem Of Calculus 12
2.1 From definite to indefinite . . . . . . . . . . . . . . . . . . . . . . . . . . 12
2.2 The theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
2.3 Some antiderivatives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17
2.4 Summary of first two chapters . . . . . . . . . . . . . . . . . . . . . . . . 21
3 Integration by substitution 22
3.1 Techniques of calculus . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22
3.2 How integration by substitution works . . . . . . . . . . . . . . . . . . . . 23
3.2.1 Method for integration by substitution . . . . . . . . . . . . . . . 24
3.3 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25
4 Trigonometry and integration 31
4.1 Previously in trigonometry... . . . . . . . . . . . . . . . . . . . . . . . . . 31
4.2 Some more antiderivatives . . . . . . . . . . . . . . . . . . . . . . . . . . 32
4.3 Using trigonometric identities . . . . . . . . . . . . . . . . . . . . . . . . 35
4.3.1 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37
1
, 4.4 Inverse functions: a different kind of substitution . . . . . . . . . . . . . . 39
4.4.1 Method for integration by trigonometric substitution . . . . . . . . 42
4.4.2 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42
5 Integration by parts 48
5.1 What is integration by parts? . . . . . . . . . . . . . . . . . . . . . . . . 48
5.2 How integration by parts works . . . . . . . . . . . . . . . . . . . . . . . 48
5.2.1 How to use integration by parts effectively . . . . . . . . . . . . . 50
5.3 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53
6 Integration using partial fractions 62
6.1 What are partial fractions? . . . . . . . . . . . . . . . . . . . . . . . . . . 62
6.2 Rational functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63
6.3 Integration using partial fractions . . . . . . . . . . . . . . . . . . . . . . 67
6.3.1 And finally... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74
2
, Chapter 1
Definite integrals
1.1 What is integration?
You may have seen that differential calculus is the study of the rate of change of quantities.
In differential calculus, the derivative f 0 (x) of a function f (x) is a function that measures
the rate of change of f (x). The derivative of a function has an application in geometry:
evaluating f 0 (x) at a point a allows you to find the gradient of the tangent to the curve of
f (x) at a.
This part of the course is concerned with integration of functions. There are two types of
integration on a function f (x):
• the indefinite integral of f (x) with respect to x, written as
Z
f (x) dx
• the definite integral of f (x) between limits a and b with respect to x, written as
Z b
f (x) dx
a
(It will be explained what this notation means shortly!)
The difference between the two types of integral is given by their uses. The indefinite integral
is precisely the reverse process of differentiation; this result is known as the Fundamental
Theorem of Calculus and is covered in Chapter 2 of the course. The definite integral
3
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