Integration by Substitution Lesson 5.5

Substitution with Indefinite Integration This is the “backwards” version of the chain rule Recall … Then …

Substitution with Indefinite Integration In general we look at the f(x) and “split” it into a g(u) and a du/dx So that …

Substitution with Indefinite Integration Note the parts of the integral from our example

Where is the 4 which we need? Example Try this … what is the g(u)? what is the du/dx? We have a problem … Where is the 4 which we need?

Where did the 1/3 come from? Example We can use one of the properties of integrals We will insert a factor of 4 inside and a factor of ¼ outside to balance the result Where did the 1/3 come from? Why is this now a 3?

Can You Tell? Which one needs substitution for integration? Go ahead and do the integration.

Try Another …

Assignment A Lesson 5.5 Page 340 Problems: 1 – 33 EOO 49 – 77 EOO

Change of Variables We completely rewrite the integral in terms of u and du Example: So u = 2x + 3 and du = 2 dx But we have an x in the integrand So we solve for x in terms of u

Change of Variables We end up with It remains to distribute the and proceed with the integration Do not forget to "un-substitute"

What About Definite Integrals Consider a variation of integral from previous slide One option is to change the limits u = 3t - 1 Then when t = 1, u = 2 when t = 2, u = 5 Resulting integral

What About Definite Integrals Also possible to "un-substitute" and use the original limits

Integration of Even & Odd Functions Recall that for an even function The function is symmetric about the y-axis Thus An odd function has The function is symmetric about the orgin

Assignment B Lesson 5.5 Page 341 Problems: 87 - 109 EOO 117 – 132 EOO