1. Integration by Inspection & Substitution
A2 Maths
Teacher: Liz

2. Example 1 Evaluate Solution:
Try and think of a function that will differentiate to
Maybe ?
Differentiate to check
We don’t want the 12 out front, so rethink:
Maybe ?
Thus,

3. Example 2 First rewrite as: Evaluate Solution:
Try and think of a function that will differentiate to
Maybe ?
Differentiate to check
We don’t want the 3/2 out front, so rethink:
Maybe ?
Thus,

4. The method we are using is called integration by inspection
The method we are using is called integration by inspection. We are using the chain rule backwards and differentiating to check our answer.

5. Example 3 – a bit more challenging
Evaluate
Solution:
Try and think of a function that will differentiate to
Maybe ?
Differentiate to check
We don’t want the 8 out front, so rethink:
Maybe ?
Thus,

6. Example 4 – a bit more challenging
We’ll come back to this one in a bit!
Evaluate
Solution:
Try and think of a function that will differentiate to
Maybe ?
Differentiate to check
There are lots of issues with this:
We don’t want the 4 out front
We need a 2x out front instead
You can continue guessing, but is there a better way?
Some problems too tricky to think through backwards. Later in this lesson, we’ll learn a method is called integration by substitution to get through those tougher problems.

7. Integration by Substitution
This is a more formal method for reversing the chain rule than inspection.
It allows to integrate much more complex looking functions.
We do this by choosing part of the integral to be replaced with a u.
We then also need to change the dx to a du to allow us to integrate with respect to u.
You will always be told what expression to use for u.

8. Integration by Substitution – Example 1
Let’s start with a simple one. Evaluate and let .
Differentiate
Solution:
We could actually work backwards and think through this one, but let’s learn the substitution method on this easier problem before moving on to something more difficult.
Rewrite integral in terms of u
Integrate
Solve for dx so we can replace it in our integral
Rewrite in terms of x

9. Integration by Substitution – Example 2
First rewrite as:
Evaluate and let
Solution:
Differentiate
Rewrite integral in terms of u
Integrate
Solve for dx so we can replace it in our integral
Rewrite in terms of x

10. Integration by Substitution – Example 3
Evaluate and let
Solution:
Differentiate
Rewrite in terms of u
We still have this 2x remaining, but it divides itself out
Solve for dx
Tidy it up and integrate
Rewrite in terms of x

11. Step by step guide Differentiate to find du/dx. Solve for dx.
Rewrite the integral in terms of u.
If there are still any remaining x’s rearrange the equation that links u and x to make x the subject and substitute in.
Integrate with respect to u.
a) If indefinite change your answer back in terms of x.
b) If definite integral change the x limits into u limits
before substituting in.

15. Copy, complete, and mark in your book. DUE November 5th!
Independent Study
Core 3 Textbook, Pg. 114, Exercise C
Copy, complete, and mark in your book. DUE November 5th!