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Trigonometric Substitution: Simplifying Integrals with Radicals

Learn how to convert radicals into cosine functions using trigonometric substitution to solve integrals effectively. Includes examples and table of trigonometric substitutions. Hyperbolic substitutions are also discussed.

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Trigonometric Substitution: Simplifying Integrals with Radicals

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  1. TRIGONOMETRIC SUBSTITUTION

  2. TRIGONOMETRIC SUBSTITUTION NOTE: We convert the radical into a cos function NOTE: We remove the radical

  3. TRIGONOMETRIC SUBSTITUTION

  4. TRIGONOMETRIC SUBSTITUTION In this section, we consider integrals containing radicals Example Find The main idea is how to get rid of the radicals. ( use trig. Substitution)

  5. TRIGONOMETRIC SUBSTITUTION

  6. TRIGONOMETRIC SUBSTITUTION

  7. TRIGONOMETRIC SUBSTITUTION Final061

  8. TRIGONOMETRIC SUBSTITUTION

  9. TRIGONOMETRIC SUBSTITUTION Final152

  10. TRIGONOMETRIC SUBSTITUTION

  11. TRIGONOMETRIC SUBSTITUTION We can transform the integrand into a function for which trigonometric substitution is appropriate by first completing the square under the root sign

  12. TRIGONOMETRIC SUBSTITUTION

  13. TRIGONOMETRIC SUBSTITUTION In the following table we list trigonometric substitutions that are effective for the given radical expressions because of the specified trigonometric identities.

  14. TRIGONOMETRIC SUBSTITUTION

  15. TRIGONOMETRIC SUBSTITUTION

  16. TRIGONOMETRIC SUBSTITUTION Example Find

  17. TRIGONOMETRIC SUBSTITUTION EXAM-2 Term-092

  18. EXAM-2Term-102

  19. TRIGONOMETRIC SUBSTITUTION Example Find

  20. TRIGONOMETRIC SUBSTITUTION Example Find hyperbolic substitutions can be used in place of trigonometric substitutions and sometimes they lead to simpler answers. But we usually use trigonometric substitutions because trigonometric identities are more familiar than hyperbolic identities.

  21. TRIGONOMETRIC SUBSTITUTION Because the ellipse is symmetric with respect to both axes, the total area is four times the area in the first quadrant

  22. TRIGONOMETRIC SUBSTITUTION

  23. TRIGONOMETRIC SUBSTITUTION

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