Lar calc10 ch05_sec2
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This document discusses techniques for integrating logarithmic and transcendental functions. It introduces the log rule for integration, which states that the integral of a rational function can be evaluated by rewriting the rational function as a logarithm. Examples show how to use long division and trigonometric identities to rewrite functions in a form suitable for the log rule. Guidelines are provided and examples given for integrating using u-substitution and properties of trigonometric integrals.
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Example 8 – Using a Trigonometric Identity
Find
Solution:
This integral does not seem to fit any formulas on our basic
list.
However, by using a trigonometric identity, you obtain
Knowing that Dx[cos x] = –sin x, you can let u = cos x and
write](https://image.slidesharecdn.com/larcalc10ch05sec2-140831021943-phpapp02/85/Lar-calc10-ch05_sec2-15-320.jpg)
Lar calc10 ch05_sec2
- 1. Logarithmic, Exponential, and Other Transcendental Functions Copyright © Cengage Learning. All rights reserved.
- 2. The Natural Logarithmic Function: Integration Copyright © Cengage Learning. All rights reserved.
- 3. 3 Objectives ■ Use the Log Rule for Integration to integrate a rational function. ■ Integrate trigonometric functions.
- 4. 4 Log Rule for Integration
- 5. 5 Log Rule for Integration The differentiation rules and produce the following integration rule.
- 6. 6 Log Rule for Integration Because the second formula can also be written as
- 7. 7 Example 1 – Using the Log Rule for Integration Because x2 cannot be negative, the absolute value notation is unnecessary in the final form of the antiderivative.
- 8. 8 Log Rule for Integration Integrals to which the Log Rule can be applied often appear in disguised form. For instance, when a rational function has a numerator of degree greater than or equal to that of the denominator, division may reveal a form to which you can apply the Log Rule. This is shown in Example 5.
- 9. 9 Example 5 – Using Long Division Before Integrating Find the indefinite integral. Solution: Begin by using long division to rewrite the integrand. Now you can integrate to obtain
- 10. 10 Example 5 – Solution Check this result by differentiating to obtain the original integrand. cont’d
- 11. 11 Log Rule for Integration The following are guidelines you can use for integration.
- 12. 12 Example 7 – u-Substitution and the Log Rule Solve the differential equation Solution: The solution can be written as an indefinite integral. Because the integrand is a quotient whose denominator is raised to the first power, you should try the Log Rule.
- 13. 13 Example 7 – Solution There are three basic choices for u. The choices u = x and u = x ln x fail to fit the u'/u form of the Log Rule. However, the third choice does fit. Letting u = lnx produces u' = 1/x, and you obtain the following. So, the solution is cont’d
- 14. 14 Integrals of Trigonometric Functions
- 15. 15 Example 8 – Using a Trigonometric Identity Find Solution: This integral does not seem to fit any formulas on our basic list. However, by using a trigonometric identity, you obtain Knowing that Dx[cos x] = –sin x, you can let u = cos x and write
- 16. 16 Example 8 – Solution cont’d
- 17. 17 Integrals of Trigonometric Functions The integrals of the six basic trigonometric functions are summarized below.