1. “Teach A Level Maths” Vol. 2: A2 Core Modules
23a: Integrating (ax+b)n
© Christine Crisp

2. Before we try to integrate compound functions, we need to be able to recognise them, and know the rule for differentiating them.
where , the inner function. If
We saw that in words this says:
differentiate the inner function
multiply by the derivative of the outer function
e.g. For
we get

3. Since indefinite integration is the reverse of differentiation, we get
So,
The rule is:
integrate the outer function

4. Since indefinite integration is the reverse of differentiation, we get
So,
The rule is:
integrate the outer function
divide by the derivative of the inner function

5. Since indefinite integration is the reverse of differentiation, we get
So,
The rule is:
integrate the outer function
If we write
divide by the derivative of the inner function
we have a clumsy “piled up” fraction so we put the 2 beside the 5.

6. Since indefinite integration is the reverse of differentiation, we get
So,
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.

7. The rule is:
integrate the outer function
divide by the derivative of the inner function
i.e. the coefficient of x
Tip: We can check the answer by differentiating it. We should get the function we wanted to integrate.
Make power one more
Drop it through the trap door
Divide by coefficient of x

8. However, we can’t integrate all compound functions in this way.
Let’s try the rule on another example:
THIS IS WRONG !
e.g.
integrate the outer function
divide by the derivative of the inner function

9. However, we can’t integrate all compound functions in this way.
Let’s try the rule on another example:
THIS IS WRONG !
e.g.
The rule has given us a quotient, which, if we differentiate it, gives:
. . . nothing like the function we wanted to integrate.

10. ? What is the important difference between and
When we differentiate the inner function of the 1st example, we get 2, a constant.
Dividing by the 2 does NOT give a quotient of the form ( since v is a function of x ).
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.

11. SUMMARY
The rule for integrating a compound function ( a function of a function ) is:
integrate the outer function
divide by 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.

12. e.g. 1.
Integrate the outer function
Divide by derivative of the inner function
i.e. the coefficient of x
( Remembering not to pile up the fractions )

13. 2. 4.