Integration by Substitution Lesson 5 5 Substitution with
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
Example • Try this … § what is the g(u)? § what is the du/dx? • We have a problem … Where is the 4 which we need?
Example • We can use one of the properties of integrals Where did the 1/3 come from? Why is this inside now anda 3? a • We will insert a factor of 4 factor of ¼ outside to balance the result
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 = 2 x + 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 = 3 t - 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 • Thus
Assignment B • Lesson 5. 5 • Page 341 • Problems: 87 - 109 EOO 117 – 132 EOO
- Slides: 15