Lesson 4.5 Integration by Substitution

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

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)

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

Example

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

Not always necessary to solve for variables.

Examples Using Substitution

Problem Set 4.5

Example: Evaluate Problem Set 4.5

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