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Similar to PPT Antiderivatives and Indefinite Integration.pptx
PPT Antiderivatives and Indefinite Integration.pptx
- 1. Review of the PreviousLesson
- 2.
- 4. JEOPARDY GAME: THE TEACHERWILL GIVE YOU THE ANSWER, THE STUDENT WILL IDENTIFY THE QUESTION.
- 5. +1 POINT -1 POINT Itsderivative is f′ x = 2𝑥.
- 6. +1 POINT -1 POINT Itsderivative is f′ x = 2𝑥. What is the derivative of 𝑓 𝑥 = 𝑥2 ?
- 7. +1 POINT -1 POINT Itsderivative is f′ x = 3𝑥2 .
- 8. +1 POINT -1 POINT Itsderivative is f′ x = 3𝑥2 . What is the derivative of 𝑓 𝑥 = 𝑥3 ?
- 9. +1 POINT -1 POINT Itsderivative is f′ x = 4𝑥.
- 10. +1 POINT -1 POINT Itsderivative is f′ x = 4𝑥. What is the derivative of 𝑓 𝑥 = 2𝑥2 ?
- 11. Its derivative isf′ x = 1 𝑥2. +2 POINTS -2 POINTS
- 12. Its derivative isf′ x = 1 𝑥2. +2 POINTS -2 POINTS What is the derivative of 𝑓 𝑥 = − 1 𝑥 ?
- 13. Its derivative isf′ x = 𝑥. +2 POINTS -2 POINTS
- 14. Its derivative isf′ x = 𝑥. +2 POINTS -2 POINTS What is the derivative of 𝑓 𝑥 = 2 3 𝑥 3 2?
- 15.
- 16. Learning Objectives: 1.Illustrate anantiderivative of a function. 2.Compute the general antiderivatives (indefinite integrals) of polynomial, radical, rational, exponential, logarithmic, and trigonometric functions.
- 17. Integral Calculus isthe branch of calculus where we study integrals and their properties. Integration is an essential concept which is the inverse process of differentiation.
- 18. Let us havean intuitive approach in finding the antiderivative of a function. Let us consider this example. 𝐟′ 𝐱 = 𝒅 𝒅𝒙 𝒙𝟐 + 𝟓 = 𝒅 𝒅𝒙 𝒙𝟐 + 𝒅 𝒅𝒙 𝟓 𝐟′ 𝐱 = 𝒅 𝒅𝒙 𝒙𝟐 + 𝟓 = 𝟐 ⋅ 𝒙𝟐−𝟏 + 𝟎 𝐟′ 𝐱 = 𝒅 𝒅𝒙 𝒙𝟐 + 𝟓 = 𝟐𝒙
- 19. First, we willadd one to the exponent of x since we subtract one from x during the process of differentiation. 𝐅 𝐱 = 𝟐𝒙𝟏+𝟏 = 𝟐𝒙𝟐
- 20. Second, we willdivide 𝐅 𝐱 = 𝟐𝒙𝟐 by its exponent 2 since we multiply the exponent during the process of differentiation. 𝐅 𝐱 = 𝟐𝒙𝟐 𝟐 𝑭 𝐱 = 𝒙𝟐
- 21. Did we alreadyrecover the original function?
- 22. The Derivative ofa Constant Let 𝑓(𝑥) be a constant function defined by 𝑦 = 𝑓(𝑥) = 𝑐, where c is a constant, then 𝒅𝒚 𝒅𝒙 = 𝒅 𝒅𝒙 𝒄 = 𝟎
- 23. We will addC (a constant arbitrary constant) to 𝐅 𝐱 = 𝒙𝟐 . (Note that the derivative of a constant is zero.) 𝐅 𝐱 = 𝒙𝟐 + 𝑪 If 𝑪 = 𝟓, then 𝐅 𝐱 = 𝒙𝟐 + 𝟓
- 24. Hence, the antiderivativeof 𝐟′ 𝐱 = 𝟐𝒙 is 𝐅 𝐱 = 𝒙𝟐 + 𝑪.
- 25. ANTIDIFFERENTIATION •This operation ofdetermining the original function from its derivative is the inverse operation of differentiation and is called antidifferentiation. •Antidifferentiation is a process or operation that reverses differentiation.
- 26. ANTIDIFFERENTIATION •Up to thispoint in Calculus, you have been concerned primarily with this problem: given a function, find its derivative •Many important applications of calculus involve the inverse problem: given the derivative of a function, find the function
- 27. The relationship betweenderivatives and antiderivatives can be represented schematically:
- 28. Definition of theAntiderivative A function 𝑭 𝒙 is called an antiderivative of a function 𝒇 on an interval 𝑰 if 𝑭′ 𝒙 = 𝒇 𝒙 for every value of 𝒙 in 𝑰.
- 29. Another term forantidifferentiation is integration. Another term for antiderivative is integral.
- 30. Integration can beclassified into two different categories, namely, •Definite Integral •Indefinite Integral
- 31. The general solutionis denoted by The expression ∫f(x)dx is read as the antiderivative of f with respect to x. So, the differential dx serves to identify x as the variable of integration. The term indefinite integral is a synonym for antiderivative. NOTATION FOR ANTIDERIVATIVES
- 32. Indefinite Integral Indefinite integralsare not defined using the upper and lower limits. The indefinite integrals represent the family of the given function whose derivatives are f, and it returns a function of the independent variable.
- 33. From our previousexample, the antiderivative of 𝒇′ 𝒙 = 𝟐𝒙 is 𝐅 𝐱 = 𝒙𝟐 + 𝑪, where C is a constant. The derivative of a constant is zero, so C can be any constant, positive or negative. The graph of 𝐅 𝐱 = 𝒙𝟐 + 𝑪 is the graph of 𝐅 𝐱 = 𝒙𝟐 shifted vertically by C units as shown in Figure 1.
- 34. Definition of theIndefinite Integral The family of antiderivatives of the function f is called the indefinite integral of f with respect to x. In symbols, this is written as 𝒇 𝒙 𝒅𝒙 Thus, if 𝐹 𝑥 is the simplest antiderivative of f and C is any arbitrary constant, then 𝒇 𝒙 𝒅𝒙 = 𝑭 𝒙 + 𝑪
- 35. The symbol ∫is just an elongated S meaning sum. This integral symbol was devised by Gottfried Wilhelm Leibniz. The dx refers to the fact that the function 𝑓 𝑥 is to be antidifferentiated or integrated with respect to the variable x. Note: ∫ 𝒇 𝒙 𝒅𝒙 is read as “the indefinite integral of 𝑓 𝑥 with respect to x”.
- 36. Definite Integral An integralthat contains the upper and lower limits (i.e.) start and end value is known as a definite integral. The value of x is restricted to lie on a real line, and a definite Integral is also called a Riemann Integral when it is bound to lie on the real line. 𝑎 𝑏 𝑓 𝑥 𝑑𝑥
- 37. Indefinite Integration Rulesof Algebraic Function
- 38. The Power Rule Ifn is any number other than −1, then 𝒙𝒏𝒅𝒙 = 𝒙𝒏+𝟏 𝒏 + 𝟏 + 𝑪 In words, when 𝑥𝑛 is integrated, the exponent n of x is increased by 1 and then 𝑥𝑛+1 is divided by the new exponent n+1. Notice that the above formula cannot be used for 𝑛 = −1.
- 39. Example 1. Evaluate∫ 𝟏𝒅𝒙. 𝒙𝟎 𝒅𝒙 𝒙𝒏 𝒅𝒙 = 𝒙𝒏+𝟏 𝒏 + 𝟏 + 𝑪 𝒙𝟎 𝒅𝒙 = 𝒙𝟎+𝟏 𝟎 + 𝟏 + 𝑪 𝒙𝟎 𝒅𝒙 = 𝒙 𝟏 + 𝑪 𝒙𝟎 𝒅𝒙 = 𝒙 + 𝑪
- 40. Example 2. Evaluate∫ 𝐱𝟓 𝐝𝐱.
- 42. Example 3. Evaluate∫ 𝟓𝒙𝒅𝒙.
- 43. Example 4. Evaluate∫ 𝟓𝒙𝟒 𝒅𝒙.
- 45. Example 5. Evaluate∫ 𝟐𝒙𝟐 + 𝟑𝒙 − 𝟒 𝒅𝒙.
- 46. Example 6. Evaluate ∫𝒙 𝟏 𝟐 − 𝒙−𝟐 + 𝟐𝝅 𝒅𝒙.
- 47. Example 7. Evaluate ∫𝒙 + 𝟑 𝒙 𝒅𝒙.
- 48. Indefinite Integration Rules ofExponential and Logarithmic Functions
- 50. Example 8. Evaluate∫ 𝟕𝒙 𝒅𝒙.
- 51. Example 9. Evaluate∫ 𝟐𝒙+𝟑 𝒅𝒙.
- 52. Example 10. Evaluate∫ 𝟗 𝒙 𝒅𝒙.
- 53. Example 11. Evaluate∫ 𝒆𝒙 − 𝟏 𝟕𝒙 𝒅𝒙.
- 54. Indefinite Integration Rules ofTrigonometric Functions
- 56. Example 12. Evaluate∫𝒔𝒊𝒏 𝒙 + 𝒄𝒐𝒔 𝒙 𝒅𝒙.
- 57. Example 13. Evaluate∫𝟒 𝐜𝐬𝐜𝟐 𝒙 − 𝟑 𝐬𝐞𝐜𝟐 𝒙 𝒅𝒙.
- 58. Example 14. Evaluate∫ 𝐭𝐚𝐧𝟐 𝐱 𝐝𝐱.
- 59. Example 15. Evaluate∫ 𝟏+𝐜𝐨𝐬𝟐 𝒙 𝐜𝐨𝐬 𝐱 𝒅𝒙
- 60. 1.∫ 𝟑𝒙 𝒅𝒙 2.∫ 𝟑𝒙+𝟑 𝒅𝒙 3.∫−𝟐 𝐜𝐨𝐬 𝒙 𝒅𝒙 4.∫ 𝟐𝟑 𝒙 𝒅𝒙 5.∫ 𝒙−𝟕 𝒅𝒙 6.∫ 𝟑𝒙 + 𝟕 𝒅𝒙 7.∫ 𝒆𝒙 − 𝟏 𝟗𝒙 𝒅𝒙 8.∫ 𝟒𝒙𝟒+𝟑𝒙𝟐+𝒙 𝒙𝟐 𝒅𝒙 9.∫ 𝟓 𝐭𝐚𝐧 𝒙 − 𝟒 𝐜𝐬𝐜𝟐 𝒙 𝒅𝒙 10.∫ 𝟏 𝟔 𝐜𝐬𝐜𝟐 𝒙 𝒅𝒙 Evaluate the following integrals below. Show your complete solution.