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Basic Concepts
of Integration



✏

✒

✑

14
...
The reverse
dx
process is to obtain the function f (x) from knowledge of its derivative
...
Applications of integration are numerous and some of these will be explored in
subsequent Blocks
...


✛

✘

Prerequisites
Before starting this Block you should
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✙

functions

Learning Style

✓ find some simple integrals by reversing
To achieve what is expected of you
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Integration as Differentiation in Reverse
dy
Suppose we differentiate the function y = x2
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Integration reverses this
2
process and we say that the integral of 2x is x
...


The situation is just a little more complicated because there are lots of functions we can differentiate to give 2x
...
5

Now do this exercise
Write down some more functions which have derivative 2x
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Consequently, when we reverse the process, we have no idea what the
original constant term might have been
...
We state that the integral of 2x is x2 + c
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In a similar way, when we want to integrate a function we use
a special notation: y(x) dx
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To integrate 2x we write


integral
sign

2x dx = x2 + c

this term is
called the
integrand

constant of integration
there must always be a
term of the form dx

Note that along with the integral sign there is a term of the form dx, which must always be
written, and which indicates the variable involved, in this case x
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The function being integrated is called the integrand
...
When you find an indefinite integral your answer should
always contain a constant of integration
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1: Integration

2

More exercises for you to try
1 a) Write down the derivatives of each of:
x3 ,

x3 + 17,

x3 − 21


b) Deduce that 3x2 dx = x3 + c
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What is meant by the term ‘integrand’ ?
3
...

Answer

2
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You could check the entries in this
table using your knowledge of differentiation
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Table of integrals
function
f (x)

indefinite integral
f (x)dx

constant, k
x
x2

kx + c
1 2
x +c
2
1 3
x +c
3
xn+1
+ c,
n+1
ln |x| + c

xn
x−1 (or
cos x
sin x
cos kx
sin kx
tan kx
ex
e−x
ekx

1
)
x

n = −1

sin x + c
− cos x + c
1
sin kx + c
k
1
− cos kx + c
k
1
ln | sec kx|+c
k
ex + c
−e−x + c
1 kx
e +c
k

When dealing
sinh x =

ex − e−x
2

cosh x =

ex + e−x
2

Note that they are simply combinations of the exponential functions ex and e−x
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Answer
Further rules for finding more complicated integrals are dealt with in subsequent Blocks
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Find  (2x − ex )dx
2
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Find  13 (x + cos 2x) dx
4
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Find (x + 3)2 dx, (be careful!)
Answer

7

Engineering Mathematics: Open Learning Unit Level 1
14
...
Computer Exercise or Activity

For this exercise it will be necessary for you to access the
computer package DERIVE
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For example to find the indefinite integral of cos 3x you would key in Author:Expression cos(3x)
followed by Calculus:Integrate
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On hitting the Simplify button DERIVE responds
SIN(3 · x)
3
Note that the constant of integration is usually omitted
...
Note that the integral
for xn is presented as
xn+1 − 1
n+1
which, up to a constant, is the correct expression
...


Engineering Mathematics: Open Learning Unit Level 1
14
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1

9

Engineering Mathematics: Open Learning Unit Level 1
14
...
g
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1
Back to the theory

Engineering Mathematics: Open Learning Unit Level 1
14
...
3x2

3x2

3x2

Back to the theory

11

Engineering Mathematics: Open Learning Unit Level 1
14
...


Back to the theory

Engineering Mathematics: Open Learning Unit Level 1
14
...

Back to the theory

13

Engineering Mathematics: Open Learning Unit Level 1
14
...


Back to the theory

Engineering Mathematics: Open Learning Unit Level 1
14
...
a) 13 t3 + c,

b) 23 x3/2 + c,
g) 14 e4x + c

c) − 12 x−2 + c,

b) 6t + c, c) − 13 cos 3t + c,

d) − 13 x−3 + c,

e) 4x + c,

d) 17 e7t + c

3
...
1: Integration

3 5
t
5

+ 23 t3/2 + c

Back to the theory

Engineering Mathematics: Open Learning Unit Level 1
14
...
Similarly

cosh x dx = sinh x + c
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1: Integration

1
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32 e2x + c

3
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− x7 + c

5
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1: Integration

18



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