table of integrals pdf

Z dx a 2+x = 1 a tan 1 x a +C 9. sec2 xdx= tanx+C 10. Numerical Integration 41 1.10. stream covers, OpenStax CNX name, and OpenStax CNX logo are not subject to the Creative Commons license and may www.mathportal.org 5. 1. Use the Print item from the "Question Help" menu to get a better print-out. © 1999-2020, Rice University. Unformatted text preview: Table of Integrals (page 2) ...Using the browser's print will lead to an undesirable print-out. Integrals of Trig. This book is Creative Commons Attribution-NonCommercial-ShareAlike License Ryzhik Alan Jeffrey, Editor University of Newcastle upon Tyne, England Daniel Zwillinger, Editor Rensselaer Polytechnic Institute, USA Translated from Russian by Scripta Technica, Inc. AMSTERDAM •BOSTON HEIDELBERG LONDON NEW YORK •OXFORD PARIS • SAN DIEGO ©2005 BE Shapiro Page 3 This document may not be reproduced, posted or published without permission. We recommend using a You could check the entries in this table using your knowledge of differentiation. S(x) dx where A(x),B(x),C(x) and D(x) are polynomials in x and S(x) is a polynomial of degree 3 or 4. double integral gives us the volume under the surface z = f(x,y), just as a single integral gives the area under a curve. ISBN 9780123736376, 9780080471112 Volumes 52 2.3. To enable us to find integrals of a wider range of functions than those normally given in a table of integrals we can make use of the following rules. Table of Standard Integrals 1. List of integrals of exponential functions 3 ( is the modified Bessel function of the first kind) References • Wolfram Mathematica Online Integrator (http:/ / integrals. Integration is the basic operation in integral calculus.While differentiation has straightforward rules by which the derivative of a complicated function can be found by differentiating its simpler component functions, integration does not, so tables of known integrals are often useful. The integral tables are extensive and quite helpful; I agree that this is a must-have for physics majors. In this section we’ve got the proof of several of the properties we saw in the Integrals Chapter as well as a couple from the Applications of Integrals Chapter. Use the Print item from the "Question Help" menu to get a better print-out. 1 of Integration by Parts 21 1.6. Table of Laplace Transforms f(t) L[f(t)] = F(s) 1 1 s (1) eatf(t) F(s a) (2) U(t a) e as s (3) f(t a)U(t a) e asF(s) (4) (t) 1 (5) (t stt 0) e 0 (6) tnf(t) ( 1)n dnF(s) dsn (7) f0(t) sF(s) f(0) (8) fn(t) snF(s) s(n 1)f(0) (fn 1)(0) (9) Z t 0 f(x)g(t x)dx F(s)G(s) (10) tn (n= 0;1;2;:::) n! sn+1 (11) tx … The result of finding an indefinite integral is usually a function plus a constant of integration. View Calc 2 TABLE OF INTEGRATION FORMULASwithlines.pdf from MATH 22 at Howard University. INTEGRAL CALCULUS - EXERCISES 42 Using the fact that the graph of f passes through the point (1,3) you get 3= 1 4 +2+2+C or C = − 5 4. Not to mention their servers gave up the ghost turned into Zombies on 25 March 2015 (Brains! Except where otherwise noted, textbooks on this site However, if the positions of several integrals coincide because they contain the same complicated function, these integrals are put in the position defined by the complexity of the second function. (3x2 + 4)d dx {u} = 12 u.u d dx { 2 − 4x2 + 7x5} = 1 2 2 − 4x2 + 7x5 (−8x + 35x4) d dx {c} = 0 , c is a constant ddx {6} = 0 , since ≅ 3.14 is a constant. View Table of integrals.pdf from MA 3253 at Mississippi State University. Unformatted text preview: Table of Integrals (page 1) ...Using the browser's print will lead to an undesirable print-out. Home Research Teaching Students CV Software Google scholar. not be reproduced without the prior and express written consent of Rice University. Purchase Table of Integrals, Series, and Products - 7th Edition. Z cscucotudu= − cscu+C 12. are licensed under a, Integration Formulas and the Net Change Theorem, Integrals Involving Exponential and Logarithmic Functions, Integrals Resulting in Inverse Trigonometric Functions, Volumes of Revolution: Cylindrical Shells, Integrals, Exponential Functions, and Logarithms, Parametric Equations and Polar Coordinates, https://openstax.org/books/calculus-volume-2/pages/1-introduction, https://openstax.org/books/calculus-volume-2/pages/a-table-of-integrals, Creative Commons Attribution 4.0 International License. Advanced Placement ® and AP ® are trademarks registered and/or owned by the College Board, which is not affiliated with, and does not endorse, this site. Table of Integrals for inverse trig forms, exponential/logarithmic forms, . sn+1 (11) tx … If the power of the sine is odd and positive: Goal: ux cos i. Partial Fractions 32 1.8. 0.2 Evaluation of double integrals To evaluate a double integral we do it in stages, starting from the inside and working out, using our knowledge of the methods for single integrals… Z secxdx= ln secx+tanx +C 12. Integration by Parts 21 1.6. Z cscudu=ln|cscu−cotu|+C 10. If the power of the sine is odd and positive: Goal: ux cos i. If you are redistributing all or part of this book in a print format, Use the Print item from the "Question Help" menu to get a better print-out. STANDARD INTEGRALS are provided. Table of Integrals, Series, and Products Seventh Edition I.S. jsp) • V. H. Moll, The Integrals in Gradshteyn and Ryzhik (http:/ / www. Table of Integrals∗ Basic Forms Integrals with Logarithms √ x ax + bdx = Z Z 1 x dx = xn+1 n+1 Z 1 dx = ln Math 22 7.5 TABLE OF INTEGRATION FORMULAS Inde…nite Integral Z Z Z Z Z Z Z Z Z Z Z Z Z Inde…nite For other types of functions, see lists of integrals First, a double integral is defined as the limit of sums. Techniques of Integration Over the next few sections we examine some techniques that are frequently successful when seeking antiderivatives of functions. As an Amazon associate we earn from qualifying purchases. ∫e du =eu +C Example 2: Evaluate ∫( ) 4x2 −5x3 +12 dx To evaluate this problem, use the first four Integral Formulas. Try this for yourself. Integration Tables - Manipulate the integrand in order to use a formula in the table of integrals. Z dx a 2+x = 1 a tan 1 x a +C 9. sec2 xdx= tanx+C 10. Trig Integrals: Integrals involving sin(x) and cos(x): Integrals involving sec(x) and tan(x): 1. /Length 3274 Provided by the Academic Center for Excellence 3 Common Derivatives and Integrals 4. , 1 1 1 + ≠− + ∫ = + C n n u u du n n 5. Partial Fractions 32 1.8. integrals are given, as well as tables of Laplace, Mellin, and other transforms, which are used in the book. Print Book & E-Book. First, a double integral is defined as the limit of sums. math. Z cosecxdx= ln cosecx cotx +C 13. Our mission is to improve educational access and learning for everyone. �L*�R#=�"4!���>�$G�VT3YcJj�\���(uT����Z��\���1ˆK�,�v����}D�R��P�����)�;��y�JH���AF��5I%�m���̧{Q���SVLTϪN�Ӫ�S��S�`� ��%�f�; �����8�Ŵ���Q{��������{D�h:�k���� ��S�ciS���+!D�r)�jw�L�d+�xr�ڸnU��%Z�v5e6�ޑ�� �����EÅ/?�� In this section we’ve got the proof of several of the properties we saw in the Integrals Chapter as well as a couple from the Applications of Integrals Chapter. This new edition contains many new results and corrections to the existing integrals. Applications of Integration 50 2.1. Z Z n+1 x 1 1 eax xn dx = , n 6= −1 dx = ln |ax Functions ∫sin cosxdx x= − ∫cos sinxdx x= − sin sin22 1 2 4 x ∫ xdx x= − cos sin22 1 2 4 x ∫ xdx x= + sin cos cos3 31 3 ∫ xdx x x= − cos sin sin3 31 3 ∫ xdx x x= − ln tan sin 2 dx x xdx x ∫ = ln tan Gradshteyn and I.M. Trigonometric Integrals and Trigonometric Substitutions 26 1.7. ISBN 9780123849335, 9780123849342 The first and second parts of the book, just as many sections, were written so that they could be read independently from each other. The first and second parts of the book, just as many sections, were written so that they could be read independently from each other. Therefore, the desired function is f(x)=1 4 In mathematics, the definite integral: ∫ is the area of the region in the xy-plane bounded by the graph of f, the x-axis, and the lines x = a and x = b, such that area above the x-axis adds to the total, and that below the x-axis subtracts from the total.. Program Boise State University Boise, Idaho, USA gradywright@boisestate.edu . html) About me. Z Z Z Z Z Z Z Z Z Table of integrals Note: Integrating constant is omitted. www.mathportal.org 5. This eighth edition of the classic Gradshteyn and Ryzhik Table of Integrals, Series, and Products is a completely revised edition of what is universally acknowledged by mathematical and applied science users as the key reference work concerning integrals and special functions. Creative Commons Attribution-NonCommercial-ShareAlike License 4.0 license. It's a pity this book is no longer in print, but it is still worth taking the trouble to buy a used copy. then you must include on every physical page the following attribution: If you are redistributing all or part of this book in a digital format, Z cotudu=ln|sinu|+C 8. A table of integrals We could use a table of derivatives to find integrals, but the more common ones are usually found in a ‘Table of Integrals’ such as that shown below. Home Research Teaching Students CV Software Google scholar. dZk� l}�����Q%PK�4�c(��^�8�pm�t�CM�à�! pdf doc ; Integration Techniques - A collection of problems using various integration techniques. Functions ∫sin cosxdx x= − ∫cos sinxdx x= − sin sin22 1 2 4 x ∫ xdx x= − cos sin22 1 2 4 x ∫ xdx x= + sin cos cos3 31 3 ∫ xdx x x= − cos sin sin3 31 3 ∫ xdx x x= − ln tan sin 2 dx x xdx x ∫ = ln tan Integrals of Trig. The integral of kkff(x) where k is a constant A constant factor in an integral can be moved outside the integral sign … This eighth edition of the classic Gradshteyn and Ryzhik Table of Integrals, Series, and Products is a completely revised edition of what is universally acknowledged by mathematical and applied science users as the key reference work concerning integrals and special functions. Elliptic integrals can be viewed as generalizations of the inverse trigonometric functions. ∫ = u +C u du ln 6. Z tanxdx= ln cosx +C 7. Integration using Tables and CAS 39 1.9. This page lists some of the most common antiderivatives Trig Integrals: Integrals involving sin(x) and cos(x): Integrals involving sec(x) and tan(x): 1. Therefore, the desired function is f(x)=1 4 MyNotesAdda.com is an online Educational Platform, where you can download free PDF for UPSC, SSC CGL, BANK, RAILWAYS, RRB NTPC, LIC AAO, and many other exams.Our Integration Formulas PDF Download (Trig, Definite, Integrals, Properties) is very Simple and Easy. A: TABLE OF BASIC DERIVATIVES Let u = u(x) be a differentiable function of the independent variable x, that is u(x) exists. Improper Integrals 46 Chapter 2. Volumes 52 2.3. CHAPTER 14 Multiple Integrals 14.1 Double Integrals 4 This chapter shows how to integrate functions of two or more variables. 7 0 obj << Section 7-5 : Proof of Various Integral Properties. Applications of Integration 50 2.1. OpenStax is part of Rice University, which is a 501(c)(3) nonprofit. Z sinxdx= cosx+C 5. For example, they can help you get Integrals containing two functions '1(x) and '2(x) are located in the division and order corresponding to the more complicated function of the two. Purchase Table of Integrals, Series, and Products - 8th Edition. Do not forget to use these tables when you need to When looking at the THEORY, STANDARD INTEGRALS, AN-SWERS or TIPS pages, use the Back button (at the bottom of the page) to return to the exercises Use the solutions intelligently. ������i�� �Q���}R|tL���3�� pz��Q洩z�*���z��>ί 4.0 and you must attribute OpenStax. This new edition contains many new results and corrections to the existing integrals. Provided by the Academic Center for Excellence 3 Common Derivatives and Integrals 4. , 1 1 1 + ≠− + ∫ = + C n n u u du n n 5. Definite Integrals 13.2 Introduction When you were first introduced to integration as the reverse of differentiation, the integrals you dealt with were indefinite integrals. 2. The integral tables are extensive and quite helpful; I agree that this is a must-have for physics majors. 1. ∫undu=un+1n+1+C,n≠−1∫undu=un+1n+1+C,n≠−1, 5. ∫sinudu=−cosu+C∫sinudu=−cosu+C, 8. ∫csc2udu=−cotu+C∫csc2udu=−cotu+C, 9. ∫secutanudu=secu+C∫secutanudu=secu+C, 10. ∫cscucotudu=−cscu+C∫cscucotudu=−cscu+C, 11. ∫tanudu=ln|secu|+C∫tanudu=ln|secu|+C, 12. ∫cotudu=ln|sinu|+C∫cotudu=ln|sinu|+C, 13. ∫secudu=ln|secu+tanu|+C∫secudu=ln|secu+tanu|+C, 14. ∫cscudu=ln|cscu−cotu|+C∫cscudu=ln|cscu−cotu|+C, 15. ∫dua2−u2=sin−1ua+C∫dua2−u2=sin−1ua+C, 16. ∫dua2+u2=1atan−1ua+C∫dua2+u2=1atan−1ua+C, 17. ∫duuu2−a2=1asec−1ua+C∫duuu2−a2=1asec−1ua+C, 18. ∫sin2udu=12u−14sin2u+C∫sin2udu=12u−14sin2u+C, 19. ∫cos2udu=12u+14sin2u+C∫cos2udu=12u+14sin2u+C, 20. ∫tan2udu=tanu−u+C∫tan2udu=tanu−u+C, 21. ∫cot2udu=−cotu−u+C∫cot2udu=−cotu−u+C, 22. ∫sin3udu=−13(2+sin2u)cosu+C∫sin3udu=−13(2+sin2u)cosu+C, 23. ∫cos3udu=13(2+cos2u)sinu+C∫cos3udu=13(2+cos2u)sinu+C, 24. ∫tan3udu=12tan2u+ln|cosu|+C∫tan3udu=12tan2u+ln|cosu|+C, 25. ∫cot3udu=−12cot2u−ln|sinu|+C∫cot3udu=−12cot2u−ln|sinu|+C, 26. ∫sec3udu=12secutanu+12ln|secu+tanu|+C∫sec3udu=12secutanu+12ln|secu+tanu|+C, 27. ∫csc3udu=−12cscucotu+12ln|cscu−cotu|+C∫csc3udu=−12cscucotu+12ln|cscu−cotu|+C, 28. ∫sinnudu=−1nsinn−1ucosu+n−1n∫sinn−2udu∫sinnudu=−1nsinn−1ucosu+n−1n∫sinn−2udu, 29. ∫cosnudu=1ncosn−1usinu+n−1n∫cosn−2udu∫cosnudu=1ncosn−1usinu+n−1n∫cosn−2udu, 30. ∫tannudu=1n−1tann−1u−∫tann−2udu∫tannudu=1n−1tann−1u−∫tann−2udu, 31. ∫cotnudu=−1n−1cotn−1u−∫cotn−2udu∫cotnudu=−1n−1cotn−1u−∫cotn−2udu, 32. ∫secnudu=1n−1tanusecn−2u+n−2n−1∫secn−2udu∫secnudu=1n−1tanusecn−2u+n−2n−1∫secn−2udu, 33. ∫cscnudu=−1n−1cotucscn−2u+n−2n−1∫cscn−2udu∫cscnudu=−1n−1cotucscn−2u+n−2n−1∫cscn−2udu, 34. ∫sinausinbudu=sin(a−b)u2(a−b)−sin(a+b)u2(a+b)+C∫sinausinbudu=sin(a−b)u2(a−b)−sin(a+b)u2(a+b)+C, 35. ∫cosaucosbudu=sin(a−b)u2(a−b)+sin(a+b)u2(a+b)+C∫cosaucosbudu=sin(a−b)u2(a−b)+sin(a+b)u2(a+b)+C, 36. ∫sinaucosbudu=−cos(a−b)u2(a−b)−cos(a+b)u2(a+b)+C∫sinaucosbudu=−cos(a−b)u2(a−b)−cos(a+b)u2(a+b)+C, 37. ∫usinudu=sinu−ucosu+C∫usinudu=sinu−ucosu+C, 38. ∫ucosudu=cosu+usinu+C∫ucosudu=cosu+usinu+C, 39. ∫unsinudu=−uncosu+n∫un−1cosudu∫unsinudu=−uncosu+n∫un−1cosudu, 40. ∫uncosudu=unsinu−n∫un−1sinudu∫uncosudu=unsinu−n∫un−1sinudu, 41. ∫sinnucosmudu=−sinn−1ucosm+1un+m+n−1n+m∫sinn−2ucosmudu=sinn+1ucosm−1un+m+m−1n+m∫sinnucosm−2udu∫sinnucosmudu=−sinn−1ucosm+1un+m+n−1n+m∫sinn−2ucosmudu=sinn+1ucosm−1un+m+m−1n+m∫sinnucosm−2udu, 42. ∫ueaudu=1a2(au−1)eau+C∫ueaudu=1a2(au−1)eau+C, 43. ∫uneaudu=1auneau−na∫un−1eaudu∫uneaudu=1auneau−na∫un−1eaudu, 44. ∫eausinbudu=eaua2+b2(asinbu−bcosbu)+C∫eausinbudu=eaua2+b2(asinbu−bcosbu)+C, 45. ∫eaucosbudu=eaua2+b2(acosbu+bsinbu)+C∫eaucosbudu=eaua2+b2(acosbu+bsinbu)+C, 46. ∫lnudu=ulnu−u+C∫lnudu=ulnu−u+C, 47. ∫unlnudu=un+1(n+1)2[(n+1)lnu−1]+C∫unlnudu=un+1(n+1)2[(n+1)lnu−1]+C, 48. ∫1ulnudu=ln|lnu|+C∫1ulnudu=ln|lnu|+C, 51. ∫tanhudu=lncoshu+C∫tanhudu=lncoshu+C, 52. ∫cothudu=ln|sinhu|+C∫cothudu=ln|sinhu|+C, 53. ∫sechudu=tan−1|sinhu|+C∫sechudu=tan−1|sinhu|+C, 54. ∫cschudu=ln|tanh12u|+C∫cschudu=ln|tanh12u|+C, 55. ∫sech2udu=tanhu+C∫sech2udu=tanhu+C, 56. ∫csch2udu=−cothu+C∫csch2udu=−cothu+C, 57. ∫sechutanhudu=−sechu+C∫sechutanhudu=−sechu+C, 58. ∫cschucothudu=−cschu+C∫cschucothudu=−cschu+C, 59. ∫sin−1udu=usin−1u+1−u2+C∫sin−1udu=usin−1u+1−u2+C, 60. ∫cos−1udu=ucos−1u−1−u2+C∫cos−1udu=ucos−1u−1−u2+C, 61. ∫tan−1udu=utan−1u−12ln(1+u2)+C∫tan−1udu=utan−1u−12ln(1+u2)+C, 62. ∫usin−1udu=2u2−14sin−1u+u1−u24+C∫usin−1udu=2u2−14sin−1u+u1−u24+C, 63. ∫ucos−1udu=2u2−14cos−1u−u1−u24+C∫ucos−1udu=2u2−14cos−1u−u1−u24+C, 64. ∫utan−1udu=u2+12tan−1u−u2+C∫utan−1udu=u2+12tan−1u−u2+C, 65. ∫unsin−1udu=1n+1[un+1sin−1u−∫un+1du1−u2],n≠−1∫unsin−1udu=1n+1[un+1sin−1u−∫un+1du1−u2],n≠−1, 66. ∫uncos−1udu=1n+1[un+1cos−1u+∫un+1du1−u2],n≠−1∫uncos−1udu=1n+1[un+1cos−1u+∫un+1du1−u2],n≠−1, 67. ∫untan−1udu=1n+1[un+1tan−1u−∫un+1du1+u2],n≠−1∫untan−1udu=1n+1[un+1tan−1u−∫un+1du1+u2],n≠−1, 68. ∫a2+u2du=u2a2+u2+a22ln(u+a2+u2)+C∫a2+u2du=u2a2+u2+a22ln(u+a2+u2)+C, 69. ∫u2a2+u2du=u8(a2+2u2)a2+u2−a48ln(u+a2+u2)+C∫u2a2+u2du=u8(a2+2u2)a2+u2−a48ln(u+a2+u2)+C, 70. ∫a2+u2udu=a2+u2−aln|a+a2+u2u|+C∫a2+u2udu=a2+u2−aln|a+a2+u2u|+C, 71. ∫a2+u2u2du=−a2+u2u+ln(u+a2+u2)+C∫a2+u2u2du=−a2+u2u+ln(u+a2+u2)+C, 72. ∫dua2+u2=ln(u+a2+u2)+C∫dua2+u2=ln(u+a2+u2)+C, 73. ∫u2dua2+u2=u2(a2+u2)−a22ln(u+a2+u2)+C∫u2dua2+u2=u2(a2+u2)−a22ln(u+a2+u2)+C, 74. ∫duua2+u2=−1aln|a2+u2+au|+C∫duua2+u2=−1aln|a2+u2+au|+C, 75. ∫duu2a2+u2=−a2+u2a2u+C∫duu2a2+u2=−a2+u2a2u+C, 76. ∫du(a2+u2)3/2=ua2a2+u2+C∫du(a2+u2)3/2=ua2a2+u2+C, 77. ∫u2−a2du=u2u2−a2−a22ln|u+u2−a2|+C∫u2−a2du=u2u2−a2−a22ln|u+u2−a2|+C, 78. ∫u2u2−a2du=u8(2u2−a2)u2−a2−a48ln|u+u2−a2|+C∫u2u2−a2du=u8(2u2−a2)u2−a2−a48ln|u+u2−a2|+C, 79. ∫u2−a2udu=u2−a2−acos−1a|u|+C∫u2−a2udu=u2−a2−acos−1a|u|+C, 80. ∫u2−a2u2du=−u2−a2u+ln|u+u2−a2|+C∫u2−a2u2du=−u2−a2u+ln|u+u2−a2|+C, 81. ∫duu2−a2=ln|u+u2−a2|+C∫duu2−a2=ln|u+u2−a2|+C, 82. ∫u2duu2−a2=u2u2−a2+a22ln|u+u2−a2|+C∫u2duu2−a2=u2u2−a2+a22ln|u+u2−a2|+C, 83. ∫duu2u2−a2=u2−a2a2u+C∫duu2u2−a2=u2−a2a2u+C, 84. ∫du(u2−a2)3/2=−ua2u2−a2+C∫du(u2−a2)3/2=−ua2u2−a2+C, 85. ∫a2−u2du=u2a2−u2+a22sin−1ua+C∫a2−u2du=u2a2−u2+a22sin−1ua+C, 86. ∫u2a2−u2du=u8(2u2−a2)a2−u2+a48sin−1ua+C∫u2a2−u2du=u8(2u2−a2)a2−u2+a48sin−1ua+C, 87. ∫a2−u2udu=a2−u2−aln|a+a2−u2u|+C∫a2−u2udu=a2−u2−aln|a+a2−u2u|+C, 88. ∫a2−u2u2du=−1ua2−u2−sin−1ua+C∫a2−u2u2du=−1ua2−u2−sin−1ua+C, 89. ∫u2dua2−u2=−u2a2−u2+a22sin−1ua+C∫u2dua2−u2=−u2a2−u2+a22sin−1ua+C, 90. ∫duua2−u2=−1aln|a+a2−u2u|+C∫duua2−u2=−1aln|a+a2−u2u|+C, 91. ∫duu2a2−u2=−1a2ua2−u2+C∫duu2a2−u2=−1a2ua2−u2+C, 92. ∫(a2−u2)3/2du=−u8(2u2−5a2)a2−u2+3a48sin−1ua+C∫(a2−u2)3/2du=−u8(2u2−5a2)a2−u2+3a48sin−1ua+C, 93. ∫du(a2−u2)3/2=ua2a2−u2+C∫du(a2−u2)3/2=ua2a2−u2+C, 94. ∫2au−u2du=u−a22au−u2+a22cos−1(a−ua)+C∫2au−u2du=u−a22au−u2+a22cos−1(a−ua)+C, 95. ∫du2au−u2=cos−1(a−ua)+C∫du2au−u2=cos−1(a−ua)+C, 96. ∫u2au−u2du=2u2−au−3a262au−u2+a32cos−1(a−ua)+C∫u2au−u2du=2u2−au−3a262au−u2+a32cos−1(a−ua)+C, 97. ∫duu2au−u2=−2au−u2au+C∫duu2au−u2=−2au−u2au+C, 98. ∫udua+bu=1b2(a+bu−aln|a+bu|)+C∫udua+bu=1b2(a+bu−aln|a+bu|)+C, 99. ∫u2dua+bu=12b3[(a+bu)2−4a(a+bu)+2a2ln|a+bu|]+C∫u2dua+bu=12b3[(a+bu)2−4a(a+bu)+2a2ln|a+bu|]+C, 100. ∫duu(a+bu)=1aln|ua+bu|+C∫duu(a+bu)=1aln|ua+bu|+C, 101. ∫duu2(a+bu)=−1au+ba2ln|a+buu|+C∫duu2(a+bu)=−1au+ba2ln|a+buu|+C, 102. ∫udu(a+bu)2=ab2(a+bu)+1b2ln|a+bu|+C∫udu(a+bu)2=ab2(a+bu)+1b2ln|a+bu|+C, 103. ∫uduu(a+bu)2=1a(a+bu)−1a2ln|a+buu|+C∫uduu(a+bu)2=1a(a+bu)−1a2ln|a+buu|+C, 104. ∫u2du(a+bu)2=1b3(a+bu−a2a+bu−2aln|a+bu|)+C∫u2du(a+bu)2=1b3(a+bu−a2a+bu−2aln|a+bu|)+C, 105. ∫ua+budu=215b2(3bu−2a)(a+bu)3/2+C∫ua+budu=215b2(3bu−2a)(a+bu)3/2+C, 106. ∫udua+bu=23b2(bu−2a)a+bu+C∫udua+bu=23b2(bu−2a)a+bu+C, 107. ∫u2dua+bu=215b3(8a2+3b2u2−4abu)a+bu+C∫u2dua+bu=215b3(8a2+3b2u2−4abu)a+bu+C, 108. ∫duua+bu=1aln|a+bu−aa+bu+a|+C,ifa>0=2−atan−1a+bu−a+C,ifa<0∫duua+bu=1aln|a+bu−aa+bu+a|+C,ifa>0=2−atan−1a+bu−a+C,ifa<0, 109. ∫a+buudu=2a+bu+a∫duua+bu∫a+buudu=2a+bu+a∫duua+bu, 110. ∫a+buu2du=−a+buu+b2∫duua+bu∫a+buu2du=−a+buu+b2∫duua+bu, 111. ∫una+budu=2b(2n+3)[un(a+bu)3/2−na∫un−1a+budu]∫una+budu=2b(2n+3)[un(a+bu)3/2−na∫un−1a+budu], 112. ∫undua+bu=2una+bub(2n+1)−2nab(2n+1)∫un−1dua+bu∫undua+bu=2una+bub(2n+1)−2nab(2n+1)∫un−1dua+bu, 113. ∫duuna+bu=−a+bua(n−1)un−1−b(2n−3)2a(n−1)∫duun−1a+bu∫duuna+bu=−a+bua(n−1)un−1−b(2n−3)2a(n−1)∫duun−1a+bu. 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