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Chapter 8

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. (1). The application of this formula is called integration by parts. In practice, we usually rewrite (1) by letting.

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Chapter 8

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  1. 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 (1) The application of this formula is called integration by parts.

  2. In practice, we usually rewrite (1) by letting This yields the following alternative form for (1)

  3. Example: Use integration by parts to evaluate Solution: Let

  4. 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. Example. Evaluate Solution: According to LIATE, we should let

  6. Example: Evaluate Solution: Let

  7. Repeated Integration by Parts Example: Evaluate Solution: Let Apply integration by parts to

  8. Finally,

  9. Example: Evaluate Solution: Let (a)

  10. Together with (a), we have Solve for the unknown integral, we have

  11. Integration by Parts for Definite Integrals For definite integrals, the formula corresponding to is

  12. Example: Evaluate Solution: Let

  13. 8.3 Trigonometric Integrals Integrating powers of sine and cosine By applying the integration by parts , we have two reduction formulas

  14. In particular, ……

  15. Integrating Products of Sines and Cosines

  16. Example: Evaluate Solution: since n=5 is odd,

  17. Integrating Powers of Tangent and Secant There are similar reduction formulas to integrate powers of tangent and secant.

  18. In particular, ……

  19. Integrating Products of Tangents and Secents

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