1. E joke
Tan x
Sin x ex
Log x

3. Lesson 4(substitution method)
Integrating
Lesson 4(substitution method)
Aims:
• To be able to integrate indefinite integrals involving substitution.
• To be able to integrate definite integrals involving substitution.
(Remember to also change the limits!)

4. Integration by substitution
The formal method of reversing the chain rule is called integration by substitution.
It is particularly useful when integrals are in one of these two forms.
Example 1: To see how this method works consider the integral
Let u = 5x + 2 so that
The problem now is that we can’t integrate a function in u with respect to x. We therefore need to write dx in terms of du.
Aside

5. Integration by substitution
Now change the variable back to x:
Example 2
Use the substitution u = 1 – 2x2 to find .
Aside
Stress that we must change the variable back to x at the end.
We can check that the final solution is correct by differentiating it using the chain rule.

6. Integration by substitution
Changing the variable back to x remembering, u = 1 – 2x2 , gives:
The final step brings the minus sign into one of the brackets to give a slightly ‘tidier’ form.

7. On w/b
Question
Use the substitution u = 4 – 3x2 to find .

8. Definite integration by substitution
When a definite integral is found by substitution it is easiest to rewrite the limits of integration in terms of the substituted variable.
Example 3: Use the substitution u = to find the area under the curve
y = between x = 1 and x = 3.
Aside

9. Definite integration by substitution
We also need to change the x limits into u limits.
As u = x then
when x = 1,
when x = 3, So
Therefore, the required area is units squared to 3SF.

10. Substitution – You try 5 minutes
Question:
Use the substitution u = x to integrate
Aside
Stress that we must change the variable back to x at the end.
We can check that the final solution is correct by differentiating it using the chain rule.
Can you see a quick way to check this?
Do exercise C page 114 and qu2 from exercise D page 115.