1. 5.7 Part I The Substitution Rule
MAT 1235 Calculus II
5.7 Part I
The Substitution Rule

2. Homework WebAssign HW 5.7 I There are 28 problems. Do it early.
This is the most common technique that you will use in the next few classes that required integration.
These problems ensure you to attain certain degree of proficiency in this topic.

3. Second Week Report – last year
Almost all of you are doing really well following instructions on how to do standard presentations.
I encourage the few of you who have not been paying attention to start doing so.
We have very limited time to grade the classwork and cannot keep wasting time to tell the few of you again and again what you need to do.

4. Preview
Antiderivatives are difficult to find. We need techniques to help us.
The substitution rule transforms a complicated integral into an easier integral.
The procedures for indefinite and definite integrals are similar but different.
Part I: Indefinite Part II Definite

5. Introductory Story
The wonderful design of windshield wipers

6. Introductory Story
The wonderful design of the integral notation…

7. The Substitution Rule for Indefinite Integrals
If 𝑢=𝑔(𝑥) is differentiable and 𝑓 is continuous on the range of 𝑢, then

8. The Substitution Rule for Indefinite Integrals
If 𝑢=𝑔(𝑥) is differentiable and 𝑓 is continuous on the range of 𝑢, then

9. Remarks The key of the sub. rule is to find the sub. 𝑢=𝑔(𝑥)
In practice, we do not memorize the formula
The design of the integral notation allows us to simplify the integral without using the formula (explicitly). For all practical purposes, we consider

10. Wonderful Design of Notation…

11. Example 1

12. Example 1

13. Example 1
You can always check the answer by differentiation:

14. Substitution Method
1.Select a substitution that appears to simplify the integrand. In particular, try to select 𝑢 so that 𝑑𝑢 is a factor in the integrand.

15. Substitution Method
1.Select a substitution that appears to simplify the integrand. In particular, try to select 𝑢 so that 𝑑𝑢 is a factor in the integrand.
2.Express the integral entirely in terms of
𝑢 and 𝑑𝑢 in one single step.

16. Substitution Method
1.Select a substitution that appears to simplify the integrand. In particular, try to select 𝑢 so that 𝑑𝑢 is a factor in the integrand.
2.Express the integral entirely in terms of
𝑢 and 𝑑𝑢 in one single step.
3.Evaluate the new (and easier) integral.
4.Express the integral in terms of the original variable.

17. Substitution Method
1.Select a substitution that appears to simplify the integrand. In particular, try to select 𝑢 so that 𝑑𝑢 is a factor in the integrand.
2.Express the integral entirely in terms of
𝑢 and 𝑑𝑢 in one single step.
3.Evaluate the new (and easier) integral.
4.Express the integral in terms of the original variable.

18. Expectations Use a two-column presentation.
Supporting info is on the right hand column. Do not interrupt the flow of the main “solution line”.
Replace all the 𝑥 by 𝑢 in one single step. Never have an integral with both variables.

19. Example 2

20. Example 3

21. Example 4

22. Example 5