Successfully reported this slideshow.
Upcoming SlideShare
×

Lesson 9 transcendental functions

6,681 views

Published on

Part of Mapua (MIT) syllabus content

Published in: Education
• Full Name
Comment goes here.

Are you sure you want to Yes No
• Thank you so much

Are you sure you want to  Yes  No

Lesson 9 transcendental functions

1. 1. TRANSCENDENTAL FUNCTIONS
2. 2. OBJECTIVES At the end of the lesson, the students are expected to: β’ use the Log Rule for Integration to integrate a rational functions. β’ integrate exponential functions. β’ integrate trigonometric functions. β’ integrate functions of the nth power of the different trigonometric functions. β’ use Walliβs Formula to shorten the solution in finding the antiderivative of powers of sine and cosine.
3. 3. β’ integrate functions whose antiderivatives involve inverse trigonometric functions. β’ use the method of completing the square to integrate a function. β’ review the basic integration rules involving elementary functions. β’ integrate hyperbolic functions. β’ integrate functions involving inverse hyperbolic functions.
4. 4. LOG RULE FOR INTEGRATION Let u be a differentiable function of x. ππ’ π’ = ππ π’ + πΆ or the above formula can also be written as π’β² π’ ππ₯ = ππ π’ + πΆ To apply this rule, look for quotients in which the numerator is the derivative of the denominator.
5. 5. β’ EXAMPLE β’ Find the indefinite integral. 1. π₯2 5βπ₯3 ππ₯ 5. π πππ₯π‘πππ₯ π πππ₯β1 ππ₯ 2. π₯3β6π₯β20 π₯+5 ππ₯ 6. π2π₯ π π₯β1 ππ₯ 3. 1 π₯πππ₯3 ππ₯ 4. 1 π₯ 2 3 1+π₯ 1 3 ππ₯
6. 6. INTEGRATION OF EXPONENTIAL FUNCTIONS Let u be a differentiable function of x. π π ππ = π π + π π π ππ = π π πππ + c
7. 7. EXAMPLE β’ Find the indefinite integral. 1. π 1 π₯ π₯2 ππ₯ 6. π₯473π₯3 ππ₯ 2. π2π₯ + πβπ₯ 2 ππ₯ 3. π₯π3π₯2+4 ππ₯ 4. 5π πππ₯2 π₯ ππ₯ 5. 10 π₯3 π₯2 ππ₯
8. 8. BASIC TRIGONOMETRIC FUNCTIONS INTEGRATION FORMULAS β’ cos π’ππ’ = sin π’ +c β’ sin π’ππ’ = -cos π’ + c β’ π ππ2 π’ππ’ = tan π’ + c β’ ππ π2 π’ππ’ = -cot π’ + π β’ sec π’ tan π’ππ’ = sec π’ + c β’ csc π’ cot π’ ππ’ = -csc π’ + c β’ tan π’ππ’ = ln sec π’ + c or - lncos π’ + c β’ cot π’ππ’ = lnsin π’ + c β’ sec π’ππ’ = ln ( sec π’ +tan π’ ) + c β’ csc π’ππ’ = -ln ( csc π’ + cot π’ ) + c
9. 9. β’ In all these formulas, u is an angle. In dealing with integrals involving trigonometric functions, transformations using the trigonometric identities are almost always necessary to reduce the integral to one or more of the standard forms.
10. 10. EXAMPLE Find the indefinite integral. 1. cos π₯ sec π₯+tan π₯ ππ₯ 2. cot 3π₯ sin 3π₯ππ₯ 7. 1βcos π₯ π ππ2 π₯ ππ₯ 3. π₯ csc π₯2 ππ₯ 8. 2 πππ 22π₯ ππ₯ 4. sin 2π₯ πππ 2 π₯ sin π₯ ππ₯ 5. cos π₯ 1β cos π₯ ππ₯ 6. (csc π₯ sin 2π₯ + 1 sin π₯ sec π₯ ) ππ₯
11. 11. TRANSFORMATION OF TRIGONOMETRIC FUNCTIONS If we are given the product of an integral power of sin π₯ and an integral power of cos π₯, where in the powers may be equal or unequal, both even, both odd, or one is even the other odd, we use the trigonometric identities and express the given integrand as a power of a trigonometric function times the derivative of that function or as the sum of powers of a function times the derivative of the function β’ We shall now see how to perform the details under specified conditions.
12. 12. POWERS OF SINE AND COSINE β’ CASE 1. πππ π ππππ π π ππ Transformations: a) If n is odd and m is even, πππ π ππππ π π = πππ πβπ ππππ π (πππ π) b) If m isoddand n is even, πππ π ππππ π π = πππ π ππππ πβπ π(ππππ) c) If n and m are both odd, transform the lesser power. If n and m are same degree either can be transformed
13. 13. CASE II. πππ π ππππ π π ππ where m and n are positive even integers. When both m and n are even, the method of type 1 fails. In this case, the identities, πππ π π = π β πππππ π , πππ π π = π+πππππ π , πππ π πππ π = πππ ππ π will be used.
14. 14. EXAMPLE β’ Evaluate the following integrals: 1. πππ 3 π₯π ππ7 π₯ ππ₯ 2. π ππ5 2π₯πππ 5 2π₯ ππ₯ 3 π ππβ3 π₯πππ 5 π₯ ππ₯ 4. π ππ2 π₯πππ 4 π₯ ππ₯ 5. π ππ4 2π₯ ππ₯ 6. πππ 2π₯ + 2π πππ₯ 2 ππ₯ 7. π ππ6 π₯πππ 4 π₯ ππ₯ 8. π ππ4 π₯πππ 5 π₯ ππ₯ 9. 0 π 2 π ππ2 π₯πππ 5 π₯ ππ₯ 10. 0 π 2 π ππ2 π₯πππ 2 π₯ ππ₯
15. 15. PRODUCT OF SINE AND COSINE β’ Integration of the products sin ππ₯ sin ππ₯ , cos ππ₯ cos ππ₯ , sin ππ₯ cos ππ₯ , where a and b are constants is carried out by using the formulas: sin π΄ sin π΅ = 1 2 cos π΄ β π΅ - 1 2 cos π΄ + π΅ sin π΄ cos π΅ = 1 2 sin π΄ β π΅ + 1 2 sin π΄ + π΅ cos π΄ cos π΅ = 1 2 cos π΄ β π΅ + 1 2 cos π΄ + π΅
16. 16. EXAMPLE β’ Perform the indicated integrations: 1. cos 8π₯ cos 5π₯ ππ₯ 2. sin 6π₯ cos 8π₯ ππ₯ 3. 2 cos 6π₯ cos β4π₯ ππ₯ 4. 2 sin(2π₯ β π) sin 3π β 2π₯ ππ₯ 5. cos 5π₯ cos 7π₯ sin 3π₯ ππ₯ 6. sin 4π₯ sin 10π₯ ππ₯ 7. 2 cos 2π₯ cos π₯ ππ₯ 8. 3 sin π₯ cos 3π₯ ππ₯
17. 17. WALLISβ FORMULA π π π πππ π ππππ π π ππ = πβ1 πβ3 ... 2 ππ 1 πβ1 πβ3 β¦ 2 ππ 1 π+π π+πβ2 β¦ 2 ππ 1 β π where in m and n are integers β₯ 0, π = π 2 , if m and n are both even, π = 1 , if either one or both are odd, and that the lower and upper limits are 0 and π 2
18. 18. EXAMPLE β’ Evaluate by Wallisβ Formula. 1. 0 π 2 π ππ4 π₯ππ₯ 2. 0 π 2 π ππ5 π₯πππ 6 π₯ππ₯ 3. 0 π 2 π ππ4 π₯πππ 8 π₯ππ₯ 4. 0 π 6 π ππ6 3π¦πππ 3 3π¦ππ¦ 5. 0 π 3 π ππ2 3π₯ 2 πππ 2 3π₯ 2 ππ₯
19. 19. POWERS OF TANGENT AND SECANT (COTANGENT AND COSECANT) I. πππ π π½ ππ½ or πππ π π½ ππ½ where n is a positive integer. When n=1 πππ π π½ ππ½= - lnπππ π½ + c πππ π π½ ππ½ =ln sin π½ + c When nβ₯ 1, we set π‘ππ π π equal to π‘ππ πβ2 π π‘ππ2 π ππ πππ‘2 π ππ¦ πππ‘ πβ2 ππππ‘2 π , replace π‘ππ2 π ππ¦ π ππ2 π β 1 ππ πππ‘2 π by (ππ π2 π β 1). Thus we get powers of tanπ and by power formula, we can evaluate the integral.
20. 20. II. πππ π π½πππ π π½ ππ½ ππ πππ π π½πππ π π½ππ½ where m and n are positive integers. β’ When m is even, we let πππ π π½ = πππ πβπ π½ πππ π π½, and express πππ πβπ π½ = (πππ π π½ + π) πβπ .We will then obtain products of powers of tan π ππ¦ π ππ2 π. The integral could be integrated by means of power formula.
21. 21. β’ If n is odd, we express πππ π πππ π π½ = πππ πβπ π½πππ πβπ π½(π¬ππ π½ π­ππ§ π½).Then we transform π‘ππ πβ1 into power of secπ using the identity πππ π π½ = πππ π π½ β π. β’ If m is odd and n is even this can be evaluated using integration by parts
22. 22. EXAMPLE β’ Find the indefinite integral. 1. π‘ππ5 π₯ππ₯ 2. π‘ππ3 π₯π ππ4 π₯ππ₯ 3. ππ π4 π₯ππ₯ 4. πππ‘2 π₯ππ π4 π₯ππ₯ 5. π‘ππ3 π₯π ππ5 π₯ππ₯
23. 23. INTEGRALS INVOLVING INVERSE TRIGONOMETRIC FUNCTIONS β’ Let u be a differentiable function of x, and let a> 0. 1. ππ’ π2βπ’2 = ππππ ππ π’ π + πΆ 2. ππ’ π2+π’2 = 1 π ππππ‘ππ π’ π + πΆ 3. ππ’ π’ π’2βπ2 = 1 π ππππ ππ π’ π + πΆ
24. 24. EXAMPLE β’ Find or evaluate the integral. 1. π₯β3 π₯2+1 ππ₯ 6. ππ2 ππ4 πβπ₯ 1βπβ2π₯ ππ₯ 2. π ππ2 π₯ 25βπ‘ππ2 π₯ ππ₯ 7. π₯ 9+8π₯2βπ₯4 ππ₯ 3. 3 2 π₯(1+π₯) ππ₯ 8. 3 6 1 25+(π₯β3)2 ππ₯ 4. 2 3 2π₯β3 4π₯βπ₯2 ππ₯ 9. π₯+2 βπ₯2β4π₯ ππ₯ 5. β2 2 ππ₯ π₯2+4π₯+13 10. 2π₯β5 π₯2+2π₯+2 ππ₯
25. 25. HYPERBOLIC FUNCTIONS β’ Definitions of the Hyperbolic Function π ππβπ₯ = ππ₯ β πβπ₯ 2 ππ πβπ₯ = 1 π ππβπ₯ , π₯ β  0 cππ βπ₯ = π π₯+πβπ₯ 2 π ππβπ₯ = 1 πππ βπ₯ π‘ππβπ₯ = π ππβπ₯ πππ βπ₯ πππ‘βπ₯ = 1 π‘ππβπ₯ , π₯ β  0
26. 26. β’ HYPERBOLIC IDENTITIES πππ β2 π’ β π ππβ2 π’ = 1 π‘ππβ2 π’ + π ππβ2 π’ = 1 πππ‘β2 π’ β ππ πβ2 π’ = 1 cosh2u = πππ β2 π’ + π ππβ2 π’ π ππβ2π₯ = 2π ππβπ₯πππ βπ₯ π ππβ2 1 2 π’ = 1 2 (πππ β π’ β 1) πππ β2 π₯ = 1 + πππ β2π₯ 2 tanh(x + y) = (π‘ππβ π₯+π‘ππβ π¦) 1+π‘ππβ π₯ π‘ππβπ¦
27. 27. INTEGRALS OF HYPERBOLIC FUNCTIONS Let u be a differentiable function of x. 1. π ππβπ’ ππ’ = πππ βπ’ + πΆ 2. πππ βπ’ ππ’ = π ππβπ’ + πΆ 3. π‘ππβπ’ ππ’ = ππ πππ βπ’ + πΆ 4. πππ‘βπ’ ππ’ = ππ π ππβ π’ + πΆ 5. π ππβπ’ ππ’ = π‘ππβ1 π ππ 6. ππ πβπ’ ππ’ = ππ(πππ‘β π’ β ππ πβπ’) + πΆ 7. π ππβ2 π’ ππ’ = π‘ππβπ’ + πΆ 8. ππ πβ2 π’ ππ’ = β πππ‘β π’ + πΆ 9. π ππβπ’ π‘ππβπ’ ππ’ = βπ ππβπ’ + πΆ 10. ππ πβ π’ πππ‘βπ’ ππ’ = βππ πβπ’ + πΆ
28. 28. INVERSE HYPERBOLIC FUNCTIONS β’ Function Domain β’ π ππββ1 π₯ = ππ(π₯ + π₯2 + 1)(ββ, +β) β’ πππ ββ1 π₯ = ππ(π₯ + π₯2 β 1) 1, β β’ π‘ππββ1 π₯ = 1 2 ππ 1+π₯ 1βπ₯ β1,1 β’ πππ‘ββ1 π₯ = 1 2 ππ π₯+1 π₯β1 ββ, β1 U(1, β) β’ π ππββ1 π₯ = ππ 1+ 1βπ₯2 π₯ (0,1 β’ ππ πββ1 π₯ = ππ 1 π₯ + 1+π₯2 π₯ ββ, 0 U(0, β)
29. 29. INTEGRATION INVOLVING INVERSE HYPERBOLIC FUNCTION β’ Let u be a differentiable function of x. β’ ππ’ π’2Β±π2 = ππ π’ + π’2 Β± π2 + πΆ β’ ππ’ π2βπ’2 = 1 2π ππ π+π’ πβπ’ + πΆ β’ ππ’ π’ π2Β±π’2 = β 1 π ππ π+ π2Β±π’2 π’ + πΆ
30. 30. EXAMPLE β’ Find the indefinite integral. 1. 1 3β9π₯2 ππ₯ 2. 1 π₯ 1+π₯ ππ₯ 3. 1 1β4π₯β2π₯2 ππ₯ 4. ππ₯ (π₯+2) π₯2+4π₯+8 5. π₯ 1+π₯3 ππ₯