“Teach A Level Maths”
Vol. 1: AS Core Modules
24: Indefinite Integration
© Christine Crisp
Indefinite Integration
Module C1
Module C2
AQA
MEI/OCR
Edexcel
OCR
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Indefinite Integration
We first need to consider an example of differentiation
e.g.1 Differentiate (a)
(a) y  x 2  3 
y  x2  3
(b) y  x  1
dy
 2x
dx
2
Equal !
(b) y  x 2  1  dy  2 x
dx
The gradient functions are the same since the graph
of y  x 2  1 is a just a translation of
y  x2  3
Indefinite Integration
Graphs of the functions
y  x2  3
e.g. the
gradient at
x = 1 is 2
y  x2  1
At each value of x, the gradients of the 2 graphs are
the same
dy
 2x
dx
Indefinite Integration
Indefinite integration is the reverse of differentiation
If we are given the gradient function and want to
find the equation of the curve, we reverse the
process of differentiation
BUT the constant is unknown
So,
dy
 2x
dx
 y  x2  C
C is called the arbitrary constant
or constant of integration
The equation y  x 2  C forms a family of curves
Indefinite Integration
e.g.2 Find the equation of the family of curves which
have a gradient function given by
Solution:
dy
 6x 2
dx
dy
 6x 2
dx
To reverse the rule of differentiation:

y  6x
3
•
•
add 1 to the power
divide by the new power
Indefinite Integration
e.g.2 Find the equation of the family of curves which
have a gradient function given by
Solution:
dy
 6x 2
dx
dy
 6x 2
dx
To reverse the rule of differentiation:
2 6x3
C

y

y  2x  C
31
3
•
•
add 1 to the power
divide by the new power
•
add C
Tip: Check the answer by differentiating
Indefinite Integration
The graphs look like this:
The gradient
function
dy
 6x 2
dx
dy
dx
dy
 6x 2
dx

y  2x3  C
( Sample of 6
values of C )
y
y  2x3  5
y  2x 3
Indefinite Integration
e.g. 3 Find the equation of the family of curves
with gradient function
dy
 3x  1
dx
Solution: The index of x in the term 3x is 1, so
adding 1 to the index gives 2.
The constant 1 has no x. It integrates to x.

3x2
y
xC
2
We can only find the value of C if we have some
additional information
Indefinite Integration
Exercises
Find the equations of the family of curves with the
following gradient functions:
1.
dy
 3x2  4x
dx
1
3 2
Ans :
3x
4x2
y

C 
21
1 3
2.
dy
1
2
 x  x 1
dx
2
3.
dy
 ( x  2)( x  3)
dx
Ans :
y  x3  2x2  C
x3 x2
y

 xC
3
4
N.B. Multiply out the
brackets first
Indefinite Integration
Exercises
Find the equations of the family of curves with the
following gradient functions:
1.
dy
 3x2  4x
dx
1
3 2
Ans :
3x
4x2
y

C 
21
1 3
2.
dy
1
2
 x  x 1
dx
2
3.
dy
 ( x  2)( x  3)
dx
Ans :
Ans :
3

2
y  x3  2x2  C
x3 x2
y

 xC
3
4
dy
 x2  x  6
dx
x
x
y

 6x  C
3
2
Indefinite Integration
Finding the value of C
e.g.1 Find the equation of the curve which passes
through the point (1, 2) and has gradient function
given by dy
2
dx
Solution:
 x x2
dy
 x2  x  2 
dx
2
x
x

y
 2x  C
2
3
3
Indefinite Integration
Finding the value of C
e.g.1 Find the equation of the curve which passes
through the point (1, 2) and has gradient function
given by dy
2
dx
Solution:
 x x2
dy
 x2  x  2 
dx
(1, 2) is on the curve:
6 is the common
denominator


2
x
x

y
 2x  C
2
3
1 1
2    2 (1)  C
3 2
3
1 1
2  2  C
3 2
1
C
6
So,

23
C
6
x3 x2
1
y

 2x 
3
2
6
Indefinite Integration
Exercises
1. Find the equation of the curve with gradient function
dy
1 which passes through the point ( 2, -2 )
 x
dx
2
2. Find the equation of the curve with gradient function
dy
 ( x  1)( x  2) which passes through the point ( 2, 1 )
dx
Indefinite Integration
Solutions
1.
dy
1
 x
dx
2
x2
y
C
4
Ans:
( 2, -2 ) lies on the curve


( 2) 2
2
C
4
 21  C

So,
1  C
2
x
y
1
4
Indefinite Integration
Solutions
2.
dy
 ( x  1)( x  2)
dx

dy
 x2  3x  2
dx
x3 3x2
 y

 2x  C
3
2
( 2, 1 ) on the curve
( 2) 3 3( 2) 2

1

 2( 2)  C
3
2
35
8
C
 1 64  C  
3
3
x3 3x2
35
So, y 

 2x 
3
2
3
Indefinite Integration
Notation for Integration
e.g. 1 We know that
dy
 2x
dx
1
2x2
 y
C
21
Another way of writing integration is:

2 x dx  x 2  C
Called the integral sign
We read this as “d x ”.
It must be included to indicate
that the variable is x
In full, we say we are integrating
“ with respect to x “.
Indefinite Integration
e.g. 2 Find (a)

 3 dt
(b)
3 dx
Solution:
(a) ( Integrate with respect to x )
 3 dx  3 x  C
 3 dt  3t  C
(b) ( Integrate with respect to t )
e.g. 3 Integrate x 3  x 2  2 x  1 with respect to x
Solution:
The notation for
integration must be
written
We have done the
integration so there
is no integral sign

x 3  x 2  2 x  1 dx
4
3
1
x
x
2x2



 xC
4
3
21
x4 x3


 x2  x  C
4
3
Indefinite Integration
Exercises
1. Find
(a)
(b)

 4t
3 x 2  4 x  2 dx
3
Ans : x 3  2 x 2  2 x  C
 8t 2  4t  3 dt
3
8
t
Ans : t 4 
 2t 2  3t  C
3
2. Integrate the following with respect to x:
(a) x 2  3 x  2
Ans : (a)
(b)
(b) 4 x 3  9 x 2  6 x  7
3
2
x
3
x
x 2  3 x  2 dx 

 2x  C
3
2

3
2
4
x

9
x
 6 x  7 dx

 x4  3x3  3x2  7x  C
Indefinite Integration
Summary
 Indefinite Integration is the reverse of differentiation.
 A constant of integration, C, is always included.
 Indefinite Integration is used to find a
family of curves.
 To find the curve through a given point, the
value of C is found by substituting for x and y.
 There are 2 notations:
dy
e.g.
 4x3  y  x4  C
dx
e.g.

4 x 3 dx  x 4  C
Indefinite Integration
Indefinite Integration
The following slides contain repeats of
information on earlier slides, shown without
colour, so that they can be printed and
photocopied.
For most purposes the slides can be printed
as “Handouts” with up to 6 slides per sheet.
Indefinite Integration
Indefinite integration is the reverse of differentiation
If we are given the gradient function and want to
find the equation of the curve, we reverse the
process of differentiation
BUT the constant is unknown
So,
dy
 2x
dx
 y  x2  C
C is called the arbitrary constant
or constant of integration
The equation y  x 2  C forms a family of curves
Indefinite Integration
e.g.
Find the equation of the family of curves which
have a gradient function given by
Solution:
dy
 6x 2
dx
dy
 6x 2
dx
To reverse the rule of differentiation:
2 6x3
C

y

y  2x  C
31
3
•
•
add 1 to the power
divide by the new power
•
add C
Tip: Check the answer by differentiating
The graphs look like this:
Indefinite Integration
The gradient
function
dy
 6x 2
dx
dy
dx
dy
 6x 2
dx
y

y  2x3  C
( Sample of 6
values of C )
y  2x3  5
y  2x 3
We can only find the value of C if we have some
additional information
Indefinite Integration
Finding the value of C
e.g.1 Find the equation of the curve which passes
through the point (1, 2) and has gradient function
given by dy
2
dx
Solution:
 x x2
dy
 x2  x  2 
dx
(1, 2) is on the curve:
6 is the common
denominator


x2
x3

y
 2x  C
2
3
1 1
2    2 (1)  C
3 2
1 1
2  2  C
3 2
1
C
6
So,

23
C
6
x3 x2
1
y

 2x 
3
2
6
Indefinite Integration
e.g. 2 Find the equation of the family of curves
with gradient function
dy
 3x  1
dx
Solution: The index of x in the term 3x is 1, so
adding 1 to the index gives 2.
The constant 1 has no x. It integrates to x.

3x2
y
xC
2
Indefinite Integration
Notation for Integration
e.g. 1 We know that
dy
 2x
dx
1
2x2
 y
C
21
Another way of writing integration is:

2 x dx  x 2  C
Called the integral sign
We read this as “d x ”.
It must be included to indicate
that the variable is x
In full, we say we are integrating
“ with respect to x “.
Indefinite Integration
Summary
 Indefinite Integration is the reverse of differentiation.
 A constant of integration, C, is always included.
 Indefinite Integration is used to find a
family of curves.
 To find the curve through a given point, the
value of C is found by substituting for x and y.
 There are 2 notations:
dy
e.g.
 4x3  y  x4  C
dx
e.g.

4 x 3 dx  x 4  C