Integration involving inverse trigonometric functions
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Lets learn a bit of involving inverse trigonometric functions. Hope you will enjoy learning it.Mainly this presentation is focused on finding inverse for trigonometric functions,we will learn right from functions to inverse functions and so the integration
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Integration involving inverse trigonometric functions
- 1. Integration involving inverse Trigonometric functions Mathematics
- 2. Inverse ⦿ Inverse exists for a one-to- one function.
- 3. One-to-one function A one-to-one function does not have two different input values that produce the same answer eg, f(x) = x - 3.
- 4. ⦿ Each of the trigonometric functions ie, Sine, cosine, tangent, secant, cosecant and cotangent has an inverse (with a restricted domain). ⦿ Let’s see them one by one
- 5. Integration ⦿ Area under the curve.
- 6. Trigonometric Functions and their inverse - •Sine Function A sine function is shown f(x) = sin(x)
- 7. Since, it fails to clear Horizontal Line Test , it’s inverse is restricted to its range.
- 8. Inverse Sine Function Inverse of Sine Function exists in – ⦿ Domain - [–1, 1] ⦿ range is [–pi/2 , pi/2].
- 9. Integration of inverse sine- Integration by parts – ⦿ Requires two functions to use Integration by parts. Let first function be 1 and second be inverse sine ie, sin−1(x) Then our integral becomes – I = ∫sin−1(x)⋅1dx
- 10. Contd… ⦿ Using the formula for integration by parts, we have ∫ {f(x) g(x) }dx = f(x) ∫g(x)dx −∫{f’(x)∫g(x)dx}dx ∫ sin−1(x) ⋅ 1 dx = sin−1(x) ∫1dx−∫ { {d/dx sin−1(x)} ∫1dx }dx
- 11. Substituting the values -
- 12. Multiplying and dividing by -2, we have - ⦿ Using formula
- 14. Hence -
- 15. ⦿ Cosine Function –
- 16. ⦿ Similar to sine, inverse of a cosine function exists in the region colored blue.
- 17. Inverse cosine Function Inverse of cosine Function exists in – ⦿ Domain - [–1, 1] ⦿ range is [0, pi].
- 18. Integration of inverse cosine- ⦿ Integration by parts – Let first function be 1 and second be inverse cosine ie, cos−1(x) Then our integral becomes – I = ∫cos−1(x)⋅1dx
- 19. ⦿ Using the formula for integration by parts, we have ∫ {f(x) g(x) }dx = f(x) ∫g(x)dx −∫{f’(x)∫g(x)dx}dx ∫ cos−1(x) ⋅ 1 dx = cos−1(x) ∫1dx−∫ { {d/dx cos−1(x)} ∫1dx }dx
- 20. ⦿ Using the formula for integration by parts, we have –
- 21. ⦿ Multiplying and dividing by -2, we have- Again using the formula -
- 23. ⦿ Hence
- 24. ⦿ Tan Function –
- 25. Inverse Tan Function Inverse of Tan Function exists in – ⦿ Domain - [Set of all Real numbers] ⦿ range is [-pi/2, pi/2].
- 26. Integration of inverse Tan- ⦿ Integration by parts – Let first function be 1 and second be inverse cosine ie, tan−1(x) Then our integral becomes –
- 27. ⦿ Using the formula for integration by parts, we have ∫ {f(x) g(x) }dx = f(x) ∫g(x)dx −∫{f’(x)∫g(x)dx}dx ∫ tan−1(x) ⋅ 1 dx = tan−1(x) ∫1dx−∫{d/dx{tan−1(x)}∫1dx}dx
- 28. ⦿ Using the formula for integration by parts, we have –
- 29. ⦿ Multiplying and dividing by -2, we have- Using the formula -
- 30. ⦿ We will get
- 31. ⦿ Hence,
- 32. ⦿ Similarly, the integration of different inverse trigonometric functions are shown below
- 33. Summary ⦿ Inverse exists for a one-to-one function. ⦿ Inverse of a function exists in its domain and
- 34. End