“Teach A Level Maths”
Vol. 2: A2 Core Modules
23: Integrating some
Compound Functions
© Christine Crisp
Integrating some Compound Functions
Module C3
AQA
Module C4
Edexcel
MEI/OCR
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Integrating some Compound Functions
Before we try to integrate compound functions, we
need to be able to recognise them, and know the
rule for differentiating them.
If
dy dy du
y  f ( g( x )) ,


dx du dx
where u  g( x ),
the inner function.
We saw that in words this says:
 differentiate the inner function
 multiply by the derivative of the outer function
e.g. For
y  sin 3 x
we get
dy
 3  cos 3 x
dx
 3 cos 3 x
Integrating some Compound Functions
Since indefinite integration is the reverse of
differentiation, we get

cos 3 x dx 

3 cos 3 x dx 
So,
The rule is:
sin 3 x  C
sin 3 x
C
3
If we divide C by 3, we get
 integrate the outer another
functionconstant, say C1, but
we usually just write C.
Integrating some Compound Functions
Since indefinite integration is the reverse of
differentiation, we get

cos 3 x dx 

3 cos 3 x dx  sin 3 x  C
So,
sin 3 x
C
3
The rule is:
 integrate the outer function
 divide by the derivative of the inner function
Integrating some Compound Functions
Since indefinite integration is the reverse of
differentiation, we get

cos 3 x dx 

3 cos 3 x dx  sin 3 x  C
So,
sin 3 x
C
3
The rule is:
 integrate the outer function
 divide by the derivative of the inner function
Integrating some Compound Functions
Since indefinite integration is the reverse of
differentiation, we get

cos 3 x dx 

3 cos 3 x dx  sin 3 x  C
So,
sin 3 x
C
3
The rule is:
 integrate the outer function
 divide by the derivative of the inner function
Tip: We can check the answer by differentiating it.
We should get the function we wanted to integrate.
Integrating some Compound Functions
However, we can’t integrate all compound functions
in this way.
Let’s try the rule on another example:
e.g.

cos x dx 
2
sin x 2  C
2x
THIS IS
WRONG !
 integrate the outer function
 divide by the derivative of the inner function
Integrating some Compound Functions
However, we can’t integrate all compound functions
in this way.
Let’s try the rule on another example:
e.g.

cos x dx 
2
sin x 2  C
2x
THIS IS
WRONG !
The rule has given us a quotient, which, if we
differentiate it, gives:
du
dv
v
 u
2
2
dx
dx  2 x( 2 x cos x )  2 (sin x )
(2 x ) 2
v2
. . . nothing like the function we wanted to integrate.
Integrating some Compound Functions
What is the important difference between
 cos 3 x dx
and
 cos x dx
2
?
When we differentiate the inner function of the 1st
example, we get 3, a constant.
Dividing by the 3 does NOT give a quotient of the
u
form
( since v is a function of x ).
v
The 2nd example gives 2x,which is a function of x.
Integrating some Compound Functions
What is the important difference between
 cos 3 x dx
and
 cos x dx
2
?
When we differentiate the inner function of the 1st
example, we get 3, a constant.
Dividing by the 3 does NOT give a quotient of the
u
form
( since v is a function of x ).
v
The 2nd example gives 2x,which is a function of x.
So, the important difference is that the 1st example
has an inner function that is linear; it differentiates
to a constant. The 2nd is non-linear.
Integrating some Compound Functions
SUMMARY
The rule for integrating a compound function ( a
function of a function ) is:
 integrate the outer function
 divide by the derivative of the inner function
provided that
the inner function is linear
 Add C
There is NO single rule for integration if the
inner function is non-linear.
Integrating some Compound Functions
Exercises
Without working them out, decide which of the
following can be integrated using the rule we have
found in this section.
1.
 sin2 x dx
2.
 (1  3 x )
3.
 2e
4.
 1 x
x
4
dx
dx
1
2
dx
5.
2
sin
 x dx
6.

7.
8.


1
dx
1  3x
1  x dx
2
1  2 x dx
We’ll now practise some integrals like 1, 2, 3, 6 and 8.
Integrating some Compound Functions
e.g. 1.
e.g. 2.


e 1 2 x
e 1 2 x
C
dx 
2
4
(1  3 x) dx 
 Integrate the outer function
Integrating some Compound Functions
e.g. 1.
e.g. 2.


e 1 2 x
e 1 2 x
C
dx 
2
5
(
1

3
x
)
(1  3 x) dx 
4
5
 Integrate the outer function
Integrating some Compound Functions
e.g. 1.
e.g. 2.


e 1 2 x
e 1 2 x
C
dx 
2
5
(
1

3
x
)
(1  3 x) dx 
C
4
5 (3)
 Integrate the outer function
 Divide by the derivative of the inner function 3.
If we write
(1  3 x ) 5
5
3
we have a clumsy “piled
up” fraction so we put
the (3 ) beside the 5.
Integrating some Compound Functions
e.g. 1.
e.g. 2.


e 1 2 x
e 1 2 x
C
dx 
2
5
(
1

3
x
)
(1  3 x) dx 
C
4
5 (3)
 Integrate the outer function
 Divide by the derivative of the inner function 3.
(1  3 x )

C
15
5
Integrating some Compound Functions
e.g. 3.

1
dx
1  3x
This is related to

1
dx  ln x  C
x
Integrate the outer function:

1
dx 
1  3x
Integrating some Compound Functions
e.g. 3.

1
dx
1  3x
This is related to

1
dx  ln x  C
x
Integrate the outer function:

ln 1  3 x
1
dx 
1  3x
Divide by the derivative of the inner function
Integrating some Compound Functions
e.g. 3.

1
dx
1  3x
This is related to

1
dx  ln x  C
x
Integrate the outer function:

ln 1  3 x
1
dx 
1  3x
3
Divide by the derivative of the inner function
Integrating some Compound Functions
e.g. 3.

1
dx
1  3x
This is related to

1
dx  ln x  C
x
Integrate the outer function:

ln 1  3 x
1
dx 
C
1  3x
3
Divide by the derivative of the inner function

ln 1  3 x
3
C
Integrating some Compound Functions
Exercises
Find
1.

sin 4 x dx
4.

2.
1
dx
1 x

2e
x
3.
dx
5.


(1  2 x ) dx
1  2 x dx
Solutions:
 cos 4 x
1. sin 4 x dx 
C

2.

4
2e
x
x
2
e
x
dx 

2
e
C
C
1
5
Integrating some Compound Functions

3.

4.
5.
6
(
1

2
x
)
(1  2 x ) dx 
C
6( 2)
(1  2 x ) 6

C
12

5
ln 1  x
1
dx 
1 x
1

1  2 x dx  (1 

(1 
1
2 x) 2
3
2 x) 2
3
 C   ln 1  x  C
dx  (1 
3
2 x) 2
3
( 2)
2
C
C
Integrating some Compound Functions
Integrating some Compound Functions
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.
Integrating some Compound Functions
SUMMARY
The rule for integrating a compound function ( a
function of a function ) is:
 integrate the outer function
 divide by the derivative of the inner function
provided that
the inner function is linear
 Add C
There is NO single rule for integration if the
inner function is non-linear.
Integrating some Compound Functions
Only the functions marked with a tick can be
integrated by the rule.
1.
2.
3.
4.

(1  3 x ) dx

2e dx

1
dx
 1 x
sin 2 x dx
4
x
2

1
6.
dx
 1  3x
7.
1  x dx

8.
1  2 x dx

5.
sin 2 x dx
2
Integrating some Compound Functions
e.g. 1.
e.g. 2.


sin 2 x dx   cos 2 x  C
2
4
(1  3 x) dx 
 Integrate the
outer function
 Divide by the
derivative of the
inner function 3.


(1  3 x )
5
5 ...
(1  3 x )
5
5 ( 3)

 ( Avoiding “piled
up” fractions )
(1  3 x ) 5

C
15
Integrating some Compound Functions
e.g. 3.

1
dx
1  3x
This is related to
So,


1
dx  ln x  C
x
ln 1  3 x
1
dx 
C
1  3x
3

ln 1  3 x
3
C