Chapter 7 – Techniques of Integration

1. Chapter 7 – Techniques of Integration 7.1 Integration by Parts 7.1 Integration by Parts

2. Introduction • Until now we have learned how to integrate by using the antiderivatives of each function. • That however will not always be the case. • Sometimes we will have to use special integration techniques to obtain indefinite integrals. 7.1 Integration by Parts

3. Introduction • In this section we will use the technique of integration by parts which corresponds to the product rule for differentiation. • For example, the product rule says: 7.1 Integration by Parts

4. Introduction • Let’s integrate both sides. • Solve for this term 7.1 Integration by Parts

5. Formula for Integration by Parts OR WHERE 7.1 Integration by Parts

6. Formula 2 Combining formula 1 with the Fundamental Theorem of Calculus we have: 7.1 Integration by Parts

7. Example 1 – Page 457 #2 Evaluate the integral using integration by parts with the indicated choices of u and dv. 7.1 Integration by Parts

8. Choosing our function u • A mnemonic device which is helpful for selecting u when using integration by parts is the LIATE principle of precedence for u: Logarithmic Inverse Trigonometric Algebraic Trigonometric Exponential • If the integrand has several factors, we try to choose let u be the highest function on the LIATE list. 7.1 Integration by Parts

9. Example 2 – Page 468 Evaluate the integral. 7.1 Integration by Parts

10. Tabular Integration Example: Find • Solution: with f (x) = x3, we list f (x) and its derivatives g(x) and its integrals Then we add the products of the functions connected by arrows, with every other sign changed, to obtain 7.1 Integration by Parts

11. Example 3 – Page 468 Evaluate the integral. 7.1 Integration by Parts

12. Find the error - 1 It’s a beautiful Spring day. You leave your calculus class feeling sad and depressed. You aren’t sad because of the class itself. On the contrary, you have just learned an amazing integration technique: Integration by Parts. You aren’t sad because it is your birthday. On the contrary, you are still young enough to actually be happy about it. You are sad because you know that every time you learn something really wonderful in calculus, a wild-eyed stranger runs up to you and shows you a “proof” that is false. Sure enough, as you cross the street, he is waiting for you on the other side. “Good morning, Kiddo,” he says. “I just learned integration by parts. Let me have it.” “What do you mean?” he asks. “Aren’t you going to run around telling me all of math is lies?” “Well, if you insist,” he chuckles…and hands you a piece of paper: 7.1 Integration by Parts

13. Find the error - continued “Hey,” you say, “I don’t get it! You did everything right this time!” “Yup!” says the hungry looking stranger. “But…Zero isn’t equal to negative one!” “Nope!” he says. You didn’t think he could pique your interest again, but he has. Spite him. Find the error in his reasoning. 7.1 Integration by Parts

14. Find the Error - 2 What a wonderful day! You have survived another encounter with the wild-eyed stranger, demolishing his mischievous pseudo-proof. As you leave his side, you can’t resist a taunt. “Didn’t your mother tell you never to forget your constants?” It seemed a better taunt when you were thinking it than it did when you said it. “Eh?” he says. You come up to him again. “I was just teasing you. Just pointing out that when doing indefinite integration, those constants should not be forgotten. A silly, simple error, not worthy of you.” You look smug. You are the victor. “Yup. Indefinite integrals always have those pesky constants.” For some reason he isn’t looking defeated. He is looking crafty. “Right. Well, I’m going to be going now…” “Of course, Kiddo, definite integrals don’t have constants, sure as elephants don’t have exoskeletons.” “Yes. Well, I really must be going.” Surprisingly quickly, he snatches the paper out of your hand, and adds to it. This is what it looks like. 7.1 Integration by Parts

15. Find the Error 2 - Continued “No constants missing here! Happy Birthday!” The stranger leaves, singing the “Happy Birthday” song in a minor key. Now there are no constants involved in the argument. But, the conclusion is the same: 0 = -1. Is the stranger right? Has he finally demonstrated that all you learned is suspect and contradictory? Or can you, using your best mathematical might, find the error in this new version of his argument? 7.1 Integration by Parts

16. Resources • Hippo Campus – Integration by Parts • Integration by Parts: The Basics – A YouTube Video • Interactive Math – Integration by Parts • Visual Calculus – Integration by Parts 7.1 Integration by Parts