1. Calculus, 8/E by Howard Anton, Irl Bivens, and Stephen Davis Copyright © 2005 by John Wiley & Sons, Inc. All rights reserved. Chapter 8 8.2 Integration By Parts Let G(x) be any antiderivative of g(x). In this case G’(x)=g(x), then by the product rule, Which implies Or, equivalently, as The application of this formula is called integration by parts. (1)

2. Calculus, 8/E by Howard Anton, Irl Bivens, and Stephen Davis Copyright © 2005 by John Wiley & Sons, Inc. All rights reserved. In practice, we usually rewrite (1) by letting This yields the following alternative form for (1)

3. Calculus, 8/E by Howard Anton, Irl Bivens, and Stephen Davis Copyright © 2005 by John Wiley & Sons, Inc. All rights reserved. Example: Use integration by parts to evaluate Solution: Let

4. Calculus, 8/E by Howard Anton, Irl Bivens, and Stephen Davis Copyright © 2005 by John Wiley & Sons, Inc. All rights reserved. Guidelines for Integration by Parts The main goal in integration by parts is to choose u and dv to obtain a new integral That is easier to evaluate than the original. A strategy that often works is to choose u and dv so that u becomes “simpler” when Differentiated, while leaving a dv that can be readily integrated to obtain v. There is another useful strategy for choosing u and dv that can be applied when the Integrand is a product of two functions from different categories in the list. Logarithmic, Inverse trigonometric, Algebraic, Trigonometric, Exponential In this case, you will often be successful if you take u to be the function whose Category occurs earlier in the list and take dv to the rest of the integrand (LIATE). This method does not work all the time, but it works often enough to b e useful.

5. Calculus, 8/E by Howard Anton, Irl Bivens, and Stephen Davis Copyright © 2005 by John Wiley & Sons, Inc. All rights reserved. Example. Evaluate Solution: According to LIATE, we should let

6. Calculus, 8/E by Howard Anton, Irl Bivens, and Stephen Davis Copyright © 2005 by John Wiley & Sons, Inc. All rights reserved. Example: Evaluate Solution: Let

7. Calculus, 8/E by Howard Anton, Irl Bivens, and Stephen Davis Copyright © 2005 by John Wiley & Sons, Inc. All rights reserved. Repeated Integration by Parts Example: Evaluate Solution: Let Apply integration by parts to

8. Calculus, 8/E by Howard Anton, Irl Bivens, and Stephen Davis Copyright © 2005 by John Wiley & Sons, Inc. All rights reserved. Finally,

9. Calculus, 8/E by Howard Anton, Irl Bivens, and Stephen Davis Copyright © 2005 by John Wiley & Sons, Inc. All rights reserved. Example: Evaluate Solution: Let (a)

10. Calculus, 8/E by Howard Anton, Irl Bivens, and Stephen Davis Copyright © 2005 by John Wiley & Sons, Inc. All rights reserved. Together with (a), we have Solve for the unknown integral, we have

11. Calculus, 8/E by Howard Anton, Irl Bivens, and Stephen Davis Copyright © 2005 by John Wiley & Sons, Inc. All rights reserved. Integration by Parts for Definite Integrals For definite integrals, the formula corresponding to is

12. Calculus, 8/E by Howard Anton, Irl Bivens, and Stephen Davis Copyright © 2005 by John Wiley & Sons, Inc. All rights reserved. Example: Evaluate Solution: Let

13. Calculus, 8/E by Howard Anton, Irl Bivens, and Stephen Davis Copyright © 2005 by John Wiley & Sons, Inc. All rights reserved. 8.3 Trigonometric Integrals Integrating powers of sine and cosine By applying the integration by parts, we have two reduction formulas

14. Calculus, 8/E by Howard Anton, Irl Bivens, and Stephen Davis Copyright © 2005 by John Wiley & Sons, Inc. All rights reserved. In particular, ……

15. Calculus, 8/E by Howard Anton, Irl Bivens, and Stephen Davis Copyright © 2005 by John Wiley & Sons, Inc. All rights reserved. Integrating Products of Sines and Cosines

16. Calculus, 8/E by Howard Anton, Irl Bivens, and Stephen Davis Copyright © 2005 by John Wiley & Sons, Inc. All rights reserved. Example: Evaluate Solution: since n=5 is odd,

17. Calculus, 8/E by Howard Anton, Irl Bivens, and Stephen Davis Copyright © 2005 by John Wiley & Sons, Inc. All rights reserved. Integrating Powers of Tangent and Secant There are similar reduction formulas to integrate powers of tangent and secant.

18. Calculus, 8/E by Howard Anton, Irl Bivens, and Stephen Davis Copyright © 2005 by John Wiley & Sons, Inc. All rights reserved. In particular, ……

19. Calculus, 8/E by Howard Anton, Irl Bivens, and Stephen Davis Copyright © 2005 by John Wiley & Sons, Inc. All rights reserved. Integrating Products of Tangents and Secents