1. Integration by Substitution

2. Integration by substitution can be used for a variety of integrals: some compound functions, some products and some quotients.
Sometimes we have a choice of method.

3. We have already met this type of integral.
See if you can do it.
Solution:
Reversing the chain rule gives
We’ll use this example to illustrate integration by substitution but if you got it right you can continue to use the reverse chain rule ( also called inspection ).

4. e.g. 1
Method:
We must substitute for x and dx.
Define u as the inner function
Let
Differentiate:
Substitute for the inner function . . .
Find dx by treating like a fraction

5. e.g. 1
Method:
We must substitute for x and dx.
Define u as the inner function
Let
Differentiate:
Substitute for the inner function . . .
and dx

6. e.g. 1
Method:
We must substitute for x and dx.
Define u as the inner function
Let
Differentiate:
Substitute for the inner function . . .
and dx

7. e.g. 1
Method:
We must substitute for x and dx.
Define u as the inner function
Let
Differentiate:
Substitute for the inner function . . .
and dx
Integrate:
Replace u:

8. e.g. 1
Method:
We must substitute for x and dx.
Define u as the inner function
Let
Differentiate:
Substitute for the inner function . . .
and dx
Integrate:
Replace u:

9. e.g. 2
This is a product but we can’t use integration by parts. Why not?
ANS: is a compound function with inner function non-linear. We can’t integrate it.
If we chose to integrate x instead, at the next stage we would have a more complicated integral than the one we started with.
We substitute as before, but using the inner function of the 2nd factor in the product.

10. Cancel the extra x e.g. 2 Define u as the inner function: Let
Differentiate:
Substitute for the inner function and dx
Cancel the extra x
If x won’t cancel we will have to make an extra substitution. We’ll do an example later.

11. So,
where
Integrate:
Substitute back:

12. SUMMARY
Substitution can be used for a variety of integrals
e.gs.
Method:
Define u as the inner function
Differentiate the substitution expression and rearrange to find dx
Substitute for the inner function and dx
If there’s an extra x, cancel it
If x won’t cancel, rearrange the substitution expression to find x and substitute for it
Integrate
Substitute back

13. Definite integration
We work in exactly the same way BUT we must also substitute for the limits, since they are values of x and we are changing the variable to u.
A definite integral gives a value so we never return to x.

14. e.g. 1
Let
Limits:
So,

15. So,

16. e.g. 2
Let
Limits:
So,

17. So,
You will often see this written as
where
We leave answers in the exact form.

18. Exercises 2. 1.
Give exact answers.

19. 1.
Solutions:
Let
Limits: So

20. 2.
Let
Limits:
So,
We can use the log laws to simplify this.