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Inverse of Trig Functions

Inverse of Trig Functions. Module 9 Lecture 3. Inverse trig functions used in eg Find the angle. sin, cos, tan. 5. 2. value. angle. . arcsin, arccos, arctan. We use the arc notation, the other common notation is. This is not a good notation as.

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Inverse of Trig Functions

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  1. Inverse of Trig Functions Module 9 Lecture 3

  2. Inverse trig functions used in eg Find the angle sin, cos, tan 5 2 value angle  arcsin, arccos, arctan We use the arc notation, the other common notation is This is not a good notation as Inverse trig functions are inverse to the normal trig functions

  3. so so Make the subject So Differentiation of Inverse Trigonometric Functions How can we differentiate arcsinx? Reduce to a function we can differentiate by setting Differentiating each side wrt x Get rid of the y

  4. Similarly for cos and tan

  5. Example 1

  6. Example 2

  7. Example 2

  8. Example 2

  9. Example 2

  10. Example 3

  11. Example 3

  12. Example 3

  13. Trigonometrical Substitutions We can make trig substitutions to integrate integrals of the following forms

  14. const = 9 coeff = 4 Example 1 1 Make substitution

  15. Example 1 1 Make substitution 2 Find dx

  16. Example 1 1 Make substitution 2 Find dx

  17. Example 1 1 Make substitution 2 Find dx

  18. dx ò 2 - 9 4 x 1 Make substitution 3 Substitute into denominator 2 Find dx

  19. dx ò 2 - 9 4 x 1 Make substitution 3 Substitute into denominator 2 Find dx

  20. dx ò 2 - 9 4 x 1 Make substitution 3 Substitute into denominator 2 Find dx

  21. dx ò 2 - 9 4 x 1 Make substitution 3 Substitute into denominator 2 Find dx

  22. dx ò 2 - 9 4 x 1 Make substitution 3 Substitute into denominator 2 Find dx

  23. dx ò 2 - 9 4 x 1 Make substitution 3 Substitute into denominator 2 Find dx 4 Evaluate integral

  24. dx ò 2 - 9 4 x 1 Make substitution 3 Substitute into denominator 2 Find dx 4 Evaluate integral

  25. dx ò 2 - 9 4 x 1 Make substitution 3 Substitute into denominator 2 Find dx 4 Evaluate integral

  26. dx ò 2 - 9 4 x 1 Make substitution 3 Substitute into denominator 2 Find dx 4 Evaluate integral

  27. dx ò 2 - 9 4 x 1 Make substitution 3 Substitute into denominator 2 Find dx 4 Evaluate integral

  28. Make sin the subject from the initial substitution 1 Make substitution 3 Substitute into denominator 2 Find dx 5 Substitute back 4 Evaluate integral

  29. dx ò 2 - 9 4 x 1 Make substitution 3 Substitute into denominator 2 Find dx 5 Substitute back 4 Evaluate integral

  30. const = 9 coeff = 4 Example 1 1 Make substitution

  31. dx ò 2 + 9 4 x 1 Make substitution 3 Substitute into denominator 2 Find dx 5 Substitute back 4 Evaluate integral

  32. dx ò 2 + 9 4 x 1 Make substitution 3 Substitute into denominator 2 Find dx 5 Substitute back 4 Evaluate integral

  33. dx ò 2 + 9 4 x 1 Make substitution 3 Substitute into denominator 2 Find dx 5 Substitute back 4 Evaluate integral

  34. dx ò 2 + 9 4 x 1 Make substitution 3 Substitute into denominator 2 Find dx 5 Substitute back 4 Evaluate integral

  35. dx ò 2 + 9 4 x 1 Make substitution 3 Substitute into denominator 2 Find dx 5 Substitute back 4 Evaluate integral

  36. dx ò 2 + 9 4 x 1 Make substitution 3 Substitute into denominator 2 Find dx 5 Substitute back 4 Evaluate integral

  37. dx ò 2 + 9 4 x 1 Make substitution 3 Substitute into denominator 2 Find dx 5 Substitute back 4 Evaluate integral

  38. dx ò 2 + 9 4 x 1 Make substitution 3 Substitute into denominator 2 Find dx 5 Substitute back 4 Evaluate integral

  39. dx ò 2 + 9 4 x 1 Make substitution 3 Substitute into denominator 2 Find dx 5 Substitute back 4 Evaluate integral

  40. dx ò 2 + 9 4 x 1 Make substitution 3 Substitute into denominator 2 Find dx 5 Substitute back 4 Evaluate integral

  41. dx ò 2 + 9 4 x 1 Make substitution 3 Substitute into denominator 2 Find dx 5 Substitute back 4 Evaluate integral

  42. dx ò 2 + 9 4 x 1 Make substitution 3 Substitute into denominator 2 Find dx 5 Substitute back 4 Evaluate integral

  43. dx ò 2 + 9 4 x 1 Make substitution 3 Substitute into denominator 2 Find dx 5 Substitute back 4 Evaluate integral

  44. dx ò 2 + 9 4 x 1 Make substitution 3 Substitute into denominator 2 Find dx 5 Substitute back 4 Evaluate integral

  45. Integrations with limits In this case we substitute for the limits and so do not need to back substitute at the end Example

  46. dx 3 / 2 ò 2 - 0 9 4 x 1 Make substitution 3 Substitute into denominator 2 Find dx Same as before 5 Evaluate integral 4 Change limits

  47. dx 3 / 2 ò 2 - 0 9 4 x 1 Make substitution 3 Substitute into denominator 2 Find dx 5 Evaluate integral 4 Change limits

  48. dx 3 / 2 ò 2 - 0 9 4 x 1 Make substitution 3 Substitute into denominator 2 Find dx 5 Evaluate integral 4 Change limits

  49. dx 3 / 2 ò 2 - 0 9 4 x 1 Make substitution 3 Substitute into denominator 2 Find dx 5 Evaluate integral 4 Change limits

  50. dx 3 / 2 ò 2 - 0 9 4 x 1 Make substitution 3 Substitute into denominator 2 Find dx 5 Evaluate integral 4 Change limits

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