Notesale: Turn your study into money

Document Preview

Extracts from the notes are below, to see the PDF you'll receive please use the links above

---

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
...


✙

functions

Learning Style

✓ find some simple integrals by reversing
To achieve what is expected of you
...
Integration as Differentiation in Reverse
dy
Suppose we differentiate the function y = x2
...
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
...
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
...
In a similar way, when we want to integrate a function we use
a special notation: y(x) dx
...
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
...
The function being integrated is called the integrand
...
When you find an indefinite integral your answer should
always contain a constant of integration
...
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
...
What is meant by the term ‘integrand’ ?
3
...

Answer

2
...
You could check the entries in this
table using your knowledge of differentiation
...


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 with the trigonometric functions the variable x must always be measured in
radians and not degrees
...
When n = −1 use the fifth entry in the table
...
1: Integration



Example Use the table above to find the indefinite integral of x7 : that is, find x7 dx

Solution



xn+1
+ c
...
With n = 7 we find

x7 dx = 18 x8 + c
From the table note that

xn dx =



Example Find the indefinite integral of cos 5x: that is, find cos 5x dx

Solution
From the table note that


cos kx dx =

With k = 5 we find

sin kx
+c
k


cos 5x dx = 15 sin 5x + c

In the table the independent variable is always given as x
...



Example Find cos 5t dt

Solution
We integrated cos 5x in the previous example
...
With k = 5 we find

cos 5t dt = 15 sin 5t + c
It follows immediately that, for example,

cos 5ω dω = 15 sin 5ω + c,


cos 5u du = 15 sin 5u + c


and so on
...


Engineering Mathematics: Open Learning Unit Level 1
14
...
You may be tempted totry to write the integrand as
n+1
x−1 and use the fourth row of the Table
...
This
 is−1because we can never divide by zero
...



Example Find 12 dx
Solution
In this example we are integrating a constant, 12
...


Now do this exercise

Find t4 dt
Answer
Now do this exercise

Find x15 dx
Use the laws of indices to write the integrand as x−5 and then use the Table
...

Use the entry in the table for integrating ekx
...
Integrate each of the following functions:
√
1/2
a) x9 ,
c) x−3 ,
d)1/x4 ,
e) 4,  f) x,
b)2x ,
2
...

3
...


g) e4x

Answer

3
...

5

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




Example Find the indefinite integral of −5 cos x; that is, find −5 cos x dx

Solution



−5 cos x dx = −5

cos x dx = −5 (sin x + c) = −5 sin x + K

where K is a constant
...
1: Integration

g(x) dx


f (x) dx −

g(x) dx

6



Example Find (x3 + sin x)dx
Solution



3

(x + sin x)dx =


3

x dx +

sin x dx = 14 x4 − cos x + c

Note that only a single constant of integration is needed
...


Answer

Now do this exercise
The hyperbolic sine and cosine functions, sinh x and cosh x are defined as follows:
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
...

Answer
Further rules for finding more complicated integrals are dealt with in subsequent Blocks
...
Find  (2x − ex )dx
2
...
Find  13 (x + cos 2x) dx
4
...
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
...


For example to find the indefinite integral of cos 3x you would key in Author:Expression cos(3x)
followed by Calculus:Integrate
...
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
...
1

9

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

1
...
32 e2x + c

3
...
− x7 + c

5
...
1: Integration

18



---