Download Maths Leaving Cert Higher Level – Integration Question 8

l–
Maths
Leaving Cert
Higher Level – Integration
Question 8 Paper 1
By Cillian Fahy
and Darron Higgins
Mocks.ie
Maths LC HL Integration © mocks.ie
Page 1
Integration Paper 1, Q8
Table of Contents:
1. Basics
2. Definite and Indefinite Integration
a. Constant of Integration
b. Definite Integrations
3. Integration by Substitution
a. Powers
b. Exponential
c. Rewriting
4. Integration and Trigonometry
a. Normal Trig. Functions
b. Using Rules for Trig. Products
c. Substitution for Trig. Functions
d. Inverse Trig. Functions
e.
5. Area
a. Curve and the x-axis
b. Two curves or a curve and a line
Mocks.ie
Maths LC HL Integration © mocks.ie
Page 2
Integration Basics:
We will begin our review of Integration with a quick overview of the basics. We will focus on
clarifying the simple points which you may have overlooked. It‟s important that you are certain on the
simple steps as that can make the more difficult questions easier.
Basics:
The basic rule of integration is: (Page 26, Table Book)
N
 x dx 
x N 1
N 1
Meaning you take the power of X add 1 to it and then divide by that new power.
Question 1: How is Integration the reverse of differentiation? The basic idea behind integration is
the following:
When you differentiate y you get
. When you integrate
you get y.
You will find that all the rules below are very similar to Differentiation.
Question 2: What if I am asked to integrate
with a number in front of it? You integrate as you
usually would without the number and then multiply by that number.
In fact, we can do the following:
 ax dx  a  x dx where a is a constant (normal number).
n
n
Question 3: What if I am asked to integrate a number without an x? Just like in Differentiation,
you think of a number without an x as a number multiplied by
because
Then just follow
the formula above.
Question 4: What if I am asked to integrate two numbers added to each other? Again, just as in
Differentiation, you integrate the terms separately and then add them together again.
Question 5: How do I deal with a Square root? You just look at the square root as
and integrate
normally then.
Question 6: How to I deal with an X below the line?
Think of what you would do for Differentiation. Just use indices to bring the X above the line and
integrate normally.
Mocks.ie
Maths LC HL Integration © mocks.ie
Page 3
Definite and Indefinite Integration:
We will now look at the two fundamental types of integrations: Definite and Indefinite. They are not
all that different but quite easy to identify.
x
N
dx This is an Indefinite Integral
a
As you can see above the
only difference is that the integration sign on the Definite Integral has a
x N dx This is a Definite Integral
number above and below
it.
b

Indefinite Integration: Constants of Integration
The Constant of Integration may sound scary but it‟s just a “+ C”. After every indefinite integration
you must add C to your answer.
You‟ll recall that we claimed that Integration was the reverse of Differentiation. Now is a good time
to really test that. Here are two questions:
y  x3 
 3x
2
dy
 3x 2
dx
 y  x3
y  x3  15 
 3x
2
dy
 3x 2
dx
 y  x3
You‟ll notice that although my beginning y values are different my final answer for y is the same.
This is clearly wrong. As differentiation gets rid of constants (like the 15 above) Integration cannot
take account of them. So instead we add a “+C” which means any number both positive and negative
could have been part of the original integration.
So the example from above should read:
y  x3  15 
 3x
2
dy
 3x 2
dx
 y  x3 + C
Mocks.ie
Maths LC HL Integration © mocks.ie
WARNING! Not including a “+C” is a
sure way to lose marks. So make
sure you include it for every
Indefinite Integration
Page 4
Definite Integration:
As you‟ve seen above you can tell a Definite Integration question from an Indefinite Integration
Question because a Definite Integration sign will have a number above and below it. But there are
other differences two:


Definite Integration does not have a “+C” ever!
With Definite Integration you will get an answer of numbers
A Definite Integration works something like this:

a
b
f ( x)dx   f (a)   f (b)
In short, you integrate, sub in the top value into that integration and then take the bottom value subbed
into the integration away from it. These values are called „limits‟.
Example: Definite Integration
 1 
1
x

2
dx
Simplify first
0
 1  2

1
x  x dx
0
1
1
 1  2 x 2  x dx
Now integrate
0
1
3


2
2

2x
x 
x  3  
2


 0
2
1
2


3
4
x
x2 
x 


3
2

 0
Substitute x  1 minus x  0
2
2
2 



3
3
4
1
1
4
0






02 
1 



 0


3
2  
3
2

 

 4 1
1  3  2    0
1
6
Mocks.ie
Maths LC HL Integration © mocks.ie
Page 5
Integration by Substitution:
Integration by Substitution is a method of integrating both Definite and Indefinite Integrals. The idea
is quite similar to Differentiation Rules where you let U and V stand for part of the sum. The
Substitution method for Integration is just slightly more advanced. Your focus is to turn complicated
sums into equations which can be computed by rules from the table book. We will look at:


Dealing with powers
Exponential Function (

Manipulation and Substitution
Example: Dealing with Powers
Leaving Cert 2006 Q8 b(i) Evaluate
 x(1  x ) dx
2 3
U = 1+x 2
du
 2x
dx
3
 (U ) xdx
U3
U4
 2 du  8
(1  x 2 )4
C
8
dx

du
 xdx
2
With powers, select whichever
part has the highest power.
Then let U equal the numbers
inside the bracket of highest
power
Sub your U value back into the final equation
WARNING! When we differentiate U above our objective is to replace dx with du. We also need to
remove all other traces of x from the question. We cannot integrate unless it is all in terms of one
variable only!
Example: Exponential Function and Substitution (
Leaving Cert 2008 Q8 b(i) Evaluate
 3x e
2 x3
dx
u
2
e
3
x
 dx
Let u = x 3
du
= 3x 2 ==> du=3x 2 dx
dx
u
u
e
du

e

WARNING! Don’t forget to sub the
value back in
e e +C
u
Mocks.ie
x3
Maths LC HL Integration © mocks.ie
Page 6
Example: Manipulation and Substitution

Integrate
x
x3
dx
x

Let u  x  3
dx
x3
x
  u  12 dx
u3
 u 
 u
u
3
3
2
1
1
2

x  u 3
du
1
dx
dx  du
du
2
 3u 
u
2
1
1
1
2
 du
2
2
3
2
 x  3 2  6 x  3  c
3
WARNING! The only element that is different here is that you needed to manipulate your U value to
get rid of the X. If you ever get to a situation where you can’t find a way to remove an X from a sum
then look closely at this.
Mocks.ie
Maths LC HL Integration © mocks.ie
Page 7
Trigonometry:
Trigonometry is a big section of the Integration Paper. Take a look at the 2010 question and you
might think it should have been on Paper 2! But don’t let that put you off, even if you are not a Trig.
Fan you’ll find the Trigonometry in Integration quite manageable as it usually follows set rules and is
mainly concerned with Sin and Cos. We will focus on:




Normal Trig. Functions
Using Rules for Products of Trig. Functions
Substitution for Trig. Functions
Inverse Trig. Functions

Normal Trigonometry Function:
Integrating Trig Functions can basically be put down to two rules:
Sin( ax )
a
Cos ( ax  l )
 Sin(ax  l )dx  
a
 Cos(ax)dx 
Or this could be seen in another way as:
, basically you differentiate whatever
the function (Cos or Sin) is acting on and divide by the result.
Leaving Cert 2008, Q8 (a) Find
 (2 x  cos3x)dx
 2 xdx   cos3xdx
x2 
{Split and Integrate separately}
sin 3x
C
3
Mocks.ie
Maths LC HL Integration © mocks.ie
Page 8
Products of Trigonometry Functions:
To integrate products of Sin and Cos, then you must follow certain formulae:
2 cos A cos B  cos( A  B)  cos( A  B)
2sin A cos B  sin( A  B)  sin( A  B)
2sin A sin B  cos( A  B)  cos( A  B)
2 cos A sin B  sin( A  B)  sin( A  B)
Table Book, page 15
WARNING! Notice in particular how important the order of Sin and Cos are in the formula.
Leaving Cert 2006 Q8, b (ii)
2sin A cos B  sin( A  B)  sin( A  B)
sin( A  B)  sin( A  B)
{Adjust the formula to suit the question}
2


sin(8 )  sin(2 )
4
sin5

cos3

d

=
d
0
04
2
sinAcosB=


sin(8 )  sin(2 )
 -cos(8 ) cos(2 )  4
d  

{Integrate as usual}
04
2
4  0
 16
 

 -cos(2 ) cos( 2 )   -cos(0) cos(0) 




4   16
4 
 16


1
1 1 1
 0  
16
16 4 4
TOP TIP: It’s always a good idea to put the larger angle first when completing these sums. It saves
you a little bit of extra work and ensures that you will not get something like sin( ).
Mocks.ie
Maths LC HL Integration © mocks.ie
Page 9
Example: Substitution for Trig Functions
These questions are quite straightforward once you know what to substitute for but if you don’t you
will find them quite difficult to deal with.
Leaving Cert 2004 Q8 b(ii)

3
0

For these questions always let U
equal the function which is to the
highest power

sin x cos3 xdx
U=cosx
(U )3 sin xdx
-du  sin xdx

3
0


3
0
U4
(U ) du = 4
3


4
(cos )
 (cos x) 4  3
(cos 0) 4
3



4  0
4
4

1 1 15
  
64 4 64
Inverse Trig. Functions:
Table Book, page 26

1
 sin 1
x
a
a2  x2
1
1
1 x

tan
 x2  a2 a
a
Integrating Inverse Trig functions is quite similar to that of Differentiation. Again you have a formula
and your focus is to manipulate your question to suit that formula.
Mocks.ie
Maths LC HL Integration © mocks.ie
Page 10
Example: Inverse Function with Manipulation


1
3
dx
81  9 x 2
dx

9(9  x )
2
dx
(3)  x
2
2
3

dx
9 x
2
{X 2 can only be multiplied by 1}
1
x
sin 1  C
3
3
Completing the Square:
These questions increase in difficulty if you are given something along the lines of:
x
2
dx
 6 x  14
This is a problem as you cannot immediately factorise this into anything workable. You then must
complete the square, as in turn this into a squared number.
x 2  6 x  13  ( x  3) 2  a
Divide the number multiplied by x
by 2 (
. This is how you know
what to square.
( x  3) 2  a  x 2  6 x  9  a  x 2  6 x  13
a  4  (2) 2
x 2  6 x  13  ( x  3) 2  (2) 2
WARNING! It’s important that both numbers are squared as that is the only way that it will fit into
the formula. If you do this method and the remainder (a) can’t be squared easily then you shouldn’t
do this method.
Mocks.ie
Maths LC HL Integration © mocks.ie
Page 11
Example: Inverse Function with Completing the
dx
 x 2  4 x  20
x 2  4 x  20  ( x  2) 2  a
Again, just divide 4 by 2 and you’ll
get 2. Therefore it must be
( x  2) 2  a  x 2  4 x  4  a  x 2  4 x  16
Square
a  16  (4) 2
x 2  4 x  20  ( x  2) 2  (4) 2
dx
1
1 x  2

tan
C
 (4)2  ( x  2)2 4
4
:
This is a nasty equation which can be quite difficult to tackle unless you know precisely what to do.
You need to always use the following substitution:
Let
x  a sin   x2  a 2 sin 2 
Example:
Integrate

3
0
9  x 2 dx
STEP ONE: Simplify
9  x 2 dx = (3) 2  x 2 dx
Let x =3sin
(3) 2  9sin 2  dx
x  a sin 
dx=-3cos d
9  9sin 2  (-3cos d )
9(1  sin 2  ) (-3cos d )
(1  sin 2  )  cos 2  Table book, p.13
9 cos 2  (-3cos d )
3cos (-3cos d ) = -9cos 2 d
Changing Limits:
Mocks.ie
Maths LC HL Integration © mocks.ie
Page 12
In this sum you are required to change your limits. This is simply because when you get to the final
part your answer is too complicated to sub x back into the equation. You original limits are in terms
of x and you want to get them in terms of . Use your original link:
to find this.
STEP TWO: Change Limits
x =3sin  3
{Use the first limit to gain a value for  }
3sin  3
sin  1
 = 90=
x=3sin  0
sin   0

2
 =0
STEP THREE: Finding the value


2
0

-9cos  d  9  2 cos 2 d {Take the -9 out}
2
0


1
9  2 cos 2 d  9  2 (1+cos2 ) d
0
0 2
Table Book, p. 14


9 2
9 
sin 2  2
(1+cos2 ) d 



0
2
2 
2  0

9 
sin 2  2
9   sin   9  sin 0 




 0
2 
2  0 2  2
2  2 
2 
9 
9
9
(  0)  (0  0) 
2 2
2
4
Mocks.ie
Maths LC HL Integration © mocks.ie
Page 13
Area:
You can be asked to find the area using integration methods. This can usually feature as a c part but
not always. There are some clear rules which if you follow should guide you through the question.


Area between the x-axis and a curve
Area between curves/curve and a line
Area, the curve and the x-axis:
b
Area   f ( x)dx
a
Where f(x) is the curve.
b, a are the points you want the area between.
Important Formulae: Not in the Table Book!
Points to watch out for:


If you get a negative value then you just take the absolute value (remove the minus). Why?
Well area cannot be negative.
If the graph drops under the other side of the x-axis, then you must calculate this area
separately. (As below)
Leaving Cert 2010 Q8 b The curve y =
crosses the x -axis at x=0, x=1 and
x=3, as shown. Calculate the total area of the shaded regions enclosed by the curve and the x-axis.
Rough Graph (Given)
Mocks.ie
Maths LC HL Integration © mocks.ie
Page 14
y = 12x 3 -48x 2 +36x
x=0, x=1,x=3
1
3
2
4
3
2


12x
-48x
+36x

3
x

16
x

18
x


1
0
0
| 3(1)4  16(1)3  18(1)2   3(0)4  16(0)3  18(0) 2  | 5

3
1
12x 3 -48x 2 +36x  3 x 4  16 x3  18 x 2 
3
1
3(3)4  16(3)3  18(3) 2   3(1) 4  16(1)3  18(1)2  
| (243  432  162)  (5) || 32 | 32m2
Area below the graph gives negative
value. Note the absolute value signs.
Area = 32+5 =37
WARNING! You can also be asked about the curve and the y-axis. It is the exact same ideas as the
question above with some slight changes (explained below). It does not feature regularly and the xaxis is a much more common question but you should still now how to approach it in case they
surprise you in the exam.
b
Area   f ( y )dx
a
Where f(y) is the curve.
b, a are the points you want the area between.
As it is on the y-axis, y=b, y=a
Important Formula: Not in the Table Book!
The important point to note about the formula above is that it is
, rather than
and
as such all your calculations will be in terms of y.
When dealing with the y-axis the rules are very similar to that of the x-axis but converted to the yaxis. Again;


If you get a negative value then you just take the absolute value (remove the minus). Why?
Well area cannot be negative.
If the graph drops under the other side of the y-axis, then you must calculate this area
separately. (As below)
Otherwise you just approach the question as you did above.
Mocks.ie
Maths LC HL Integration © mocks.ie
Page 15
Area between Curves/Curve and a Line:
b
b
a
a
Area=  f ( x)dx   g ( x)dx
Where f(x), g(x) are the two curves/curve and a line
b, a are the points where they intersect
This is the more difficult type of question as you need to find a lot of the points yourself. Again the
same rules apply in that area must be positive, therefore negative values become positives ones and
that area below the axis should be calculated separately from area above it.
Leaving Cert 2008, Q8 c The diagram shows the curve y = 4
and the line 2x
= 0.
Calculate the area of the shaded region enclosed by the curve and the line
Mocks.ie
Maths LC HL Integration © mocks.ie
Page 16
STEP ONE: Cuts x-axis
2x+y-1=0
y=0
{Cuts the x-axis}
2x-1=0
x=
1
2
y  4  x2
0  4  x2
y=0 {Again, cuts x-axis}
x2  4
x  2
x  2,
{Judge this by the graph}
STEP TWO: Find intersection points
y=x 2  4
2x  y 1  0
y  -2 x +1
2 x  1  4  x
x2  2x  3  0
(x-3)(x+1)=0
Mocks.ie
2
x=3, x=-1
Maths LC HL Integration © mocks.ie
Page 17
STEP THREE: Area above x-axis

2
1
1
2
1
4  x dx   1  2 x 
2

x3 
4 x  3 


2

x3 
4 x  3 


2
  x  x 
2
1
1
2
1
8
1 16 11
 (8  )  (4  )    9
3
3
3 3
1
1
2
1 1
1
 (  )  (1  1)  2
1
2 4
4
1
3
92  6
4
4
 x  x 
2
STEP FOUR: Area below x-axis
3
3
2
1

2
xdx

4

x
dx 
1


2
2
3
2 3
 x  x 
 x  x 2 
1
2
3
1
2

x3 
 4 x   
32

1 1
1
 (3  9)  (  )  6
2 4
4
WARNING! Absolute value signs
should not go in here. They are
only put in when you are adding
these numbers together in the
last line!
3

x3 
8 9 16
7
4
x


(12

9)

(8

)





3  2
3 3 3
3

1
7
75 28
47 47
| (6 )  ( ) |   ||  |
4
3
12 12
12 12
STEP FIVE: Add Area together
3 47 81 47 128
6 
 

4 12 12 12 12
Mocks.ie
{WARNING! This is the step that is easy forget. Don't!}
Maths LC HL Integration © mocks.ie
Page 18
Mocks.ie
Maths LC HL Integration © mocks.ie
Page 19