# PPT Antiderivatives and Indefinite Integration.pptx

30 May 2023                                                            1 sur 60

### PPT Antiderivatives and Indefinite Integration.pptx

• 1. Review of the Previous Lesson
• 2. ACTIVITY
• 4. JEOPARDY GAME: THE TEACHER WILL GIVE YOU THE ANSWER, THE STUDENT WILL IDENTIFY THE QUESTION.
• 5. +1 POINT -1 POINT Its derivative is f′ x = 2𝑥.
• 6. +1 POINT -1 POINT Its derivative is f′ x = 2𝑥. What is the derivative of 𝑓 𝑥 = 𝑥2 ?
• 7. +1 POINT -1 POINT Its derivative is f′ x = 3𝑥2 .
• 8. +1 POINT -1 POINT Its derivative is f′ x = 3𝑥2 . What is the derivative of 𝑓 𝑥 = 𝑥3 ?
• 9. +1 POINT -1 POINT Its derivative is f′ x = 4𝑥.
• 10. +1 POINT -1 POINT Its derivative is f′ x = 4𝑥. What is the derivative of 𝑓 𝑥 = 2𝑥2 ?
• 11. Its derivative is f′ x = 1 𝑥2. +2 POINTS -2 POINTS
• 12. Its derivative is f′ x = 1 𝑥2. +2 POINTS -2 POINTS What is the derivative of 𝑓 𝑥 = − 1 𝑥 ?
• 13. Its derivative is f′ x = 𝑥. +2 POINTS -2 POINTS
• 14. Its derivative is f′ x = 𝑥. +2 POINTS -2 POINTS What is the derivative of 𝑓 𝑥 = 2 3 𝑥 3 2?
• 15. Antiderivatives and Indefinite Integration
• 16. Learning Objectives: 1.Illustrate an antiderivative of a function. 2.Compute the general antiderivatives (indefinite integrals) of polynomial, radical, rational, exponential, logarithmic, and trigonometric functions.
• 17. Integral Calculus is the 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 have an 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 will add 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 will divide 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 already recover the original function?
• 22. The Derivative of a 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 add C (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 antiderivative of 𝐟′ 𝐱 = 𝟐𝒙 is 𝐅 𝐱 = 𝒙𝟐 + 𝑪.
• 25. ANTIDIFFERENTIATION •This operation of determining 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 this point 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 between derivatives and antiderivatives can be represented schematically:
• 28. Definition of the Antiderivative 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 for antidifferentiation is integration. Another term for antiderivative is integral.
• 30. Integration can be classified into two different categories, namely, •Definite Integral •Indefinite Integral
• 31. The general solution is 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 integrals are 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 previous example, 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 the Indefinite 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 integral that 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 Rules of Algebraic Function
• 38. The Power Rule If n 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 of Exponential 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 of Trigonometric 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.