1. Lesson 4.5 Integration by Substitution

2. Recognizing the “Outside-Inside” Pattern
From doing derivatives we need to recognize the integrand above is a composite function from the “derivative of the outside times the derivative of the inside” (chain rule).
“+ C” since this is an indefinite integral

3. f(g(x)) g’(x) Think of this function as 2 functions: f(x) and g(x)
As a composite function then:
Now look at the original integral:
outside
inside
f(g(x))
g’(x)

4. Read this as “the antiderivative of the outside function with the inside function plugged in…plus C”

6. Example

9. Leibniz Notation Let u = x4 + 2
Changing Variables (“Chain Rule’s Revenge”)
We learned to think so we didn’t have to really use the chain rule
Now it will become useful again:
Leibniz Notation
Differentiate to sub for dx
Use this to make any substitutions for x
Let u = x4 + 2

11. Not always necessary to solve for variables.

13. Examples Using Substitution

14. Problem Set 4.5

15. Example: Evaluate
Problem Set 4.5

16. Prove its symmetry: Remember f(-x) = -f(x) means odd
And f(-x) = f(x) means even.