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### table of integrals pdf

/Length 3274 2. Z eu du = eu +C 4. Z cosxdx= sinx+C 6. Z cosudu= sinu+C 6. 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. "�0+\�� {Z'�˰�������-�������Ch�X��p aN���� �pr"��ʿ�)����C�i�٣&g[e=&���O%�D��:�V��+^�� �qH9&���*�J0E@��2mU�g:��8�fng��W�����K��|�%Qi���aHmS��VU�� not be reproduced without the prior and express written consent of Rice University. We recommend using a 1 of Arc Length, Parametric Curves 57 2.4. Z sinudu= − cosu+C 5. Z e xdx= e +C 4. ������i�� �Q���}R|tL���3�� pz��Q洩z�*���z��>ί 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. About me. stream Unformatted text preview: Table of Integrals (page 1) ...Using the browser's print will lead to an undesirable print-out. Integration by Parts 21 1.6. Save a du x dx sin( ) ii. Z e xdx= e +C 4. 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. 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 You could check the entries in this table using your knowledge of diﬀerentiation. Z dx a 2+x = 1 a tan 1 x a +C 9. sec2 xdx= tanx+C 10. 1.5. We also Cover Basic Topics like Maths, Geography, H istory, Polity, etc and study materials including previous Year … The clustrmap is periodically (and automatically) archived and its counters reset, so the total is smaller. Second, we find a fast way to compute it. Elliptic integrals can be viewed as generalizations of the inverse trigonometric functions. Â© 1999-2020, Rice University. Program Boise State University Boise, Idaho, USA gradywright@boisestate.edu . %���� Table of Integrals∗ Basic Forms Integrals with Logarithms √ x ax + bdx = Z Z 1 x dx = xn+1 n+1 Z 1 dx = ln Purchase Table of Integrals, Series, and Products - 7th Edition. Home Research Teaching Students CV Software Google scholar. Integrals of Trig. This page lists some of the most common antiderivatives Applications of Integration 50 2.1. Therefore, the desired function is f(x)=1 4 A: TABLE OF BASIC DERIVATIVES Let u = u(x) be a differentiable function of the independent variable x, that is u(x) exists. Its full title today is Table of Integrals, Series, and Products.. Integration Tables - Manipulate the integrand in order to use a formula in the table of integrals. CHAPTER 14 Multiple Integrals 14.1 Double Integrals 4 This chapter shows how to integrate functions of two or more variables. The following is a list of integrals (antiderivative functions) of rational functions.Any rational function can be integrated by partial fraction decomposition of the function into a sum of functions of the form: (−), and + ((−) +).which can then be integrated term by term. 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… For other types of functions, see lists of integrals >> This new edition contains many new results and corrections to the existing integrals. Print Book & E-Book. Z Z n+1 x 1 1 eax xn dx = , n 6= −1 dx = ln |ax CHAPTER 14 Multiple Integrals 14.1 Double Integrals 4 This chapter shows how to integrate functions of two or more variables. Integration using Tables and CAS 39 1.9. 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. For example, faced with Z x10 dx First, a double integral is defined as the limit of sums. u ddx {(x3 + 4x + 1)3/4} = 34 (x3 + 4x + 1)−1/4. The key idea is to replace a double integral by two ordinary "single" integrals. View Table of integrals.pdf from MA 3253 at Mississippi State University. 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. www.mathportal.org 5. Use the Print item from the "Question Help" menu to get a better print-out. Professor Department of Mathematics Computing Ph.D. ∫ = u +C u du ln 6. pdf doc ; Integration Techniques - A collection of problems using various integration techniques. Z dx a 2+x = 1 a tan 1 x a +C 9. sec2 xdx= tanx+C 10. 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. Except where otherwise noted, textbooks on this site This book is Creative Commons Attribution-NonCommercial-ShareAlike License ∫e du =eu +C Example 2: Evaluate ∫( ) 4x2 −5x3 +12 dx To evaluate this problem, use the first four Integral Formulas. Z sinxdx= cosx+C 5. View integral tables.pdf from MATH 220 at University of Missouri, Kansas City. 1 of Unformatted text preview: Table of Integrals (page 2) ...Using the browser's print will lead to an undesirable print-out. Creative Commons Attribution-NonCommercial-ShareAlike License 4.0 license. Improper Integrals 46 Chapter 2. Trig Integrals: Integrals involving sin(x) and cos(x): Integrals involving sec(x) and tan(x): 1. 1. 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. The result of ﬁnding an indeﬁnite integral is usually a function plus a constant of integration. ISBN 9780123736376, 9780080471112 Z secxdx= ln secx+tanx +C 12. Volumes 52 2.3. �L*�R#=�"4!���>�$G�VT3YcJj�\���(uT����Z��\���1ˆK�,�v����}D�R��P�����)�;��y�JH���AF��5I%�m���̧{Q���SVLTϪN�Ӫ�S��S�� ��%�f�; �����8�Ŵ���Q{��������{D�h:�k���� ��S�ciS���+!D�r)�jw�L�d+�xr�ڸnU��%Z�v5e6�ޑ�� �����EÅ/?�� STANDARD INTEGRALS are provided. Table of Integrals, Series, and Products Seventh Edition I.S. Z Z Z Z Z Z Z Z Z Table of integrals Note: Integrating constant is omitted. The ﬁrst and second parts of the book, just as many sections, were written so that they could be read independently from each other. Partial Fractions 32 1.8. In mathematics, the definite integral: ∫ is the area of the region in the xy-plane bounded by the graph of f, the x-axis, and the lines x = a and x = b, such that area above the x-axis adds to the total, and that below the x-axis subtracts from the total.. More about Areas 50 2.2. Ryzhik Alan Jeffrey, Editor University of Newcastle upon Tyne, England Daniel Zwillinger, Editor Rensselaer Polytechnic Institute, USA Translated from Russian by Scripta Technica, Inc. AMSTERDAM •BOSTON HEIDELBERG LONDON NEW YORK •OXFORD PARIS • SAN DIEGO #ӻ�D�"$Ӧ�W��6dKa��e�π;N�2i����~�8�ϙ�.� ��AF1�"��;{��gW��ˌ3$|C�h����f����@xC$ Use the Print item from the "Question Help" menu to get a better print-out. Z cosecxdx= ln cosecx cotx +C 13. Z secutanudu= secu+C 11. Gradshteyn and I.M. 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 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. ® and AP ® are trademarks registered and/or owned by the College Board, which is not affiliated with, and does not endorse, this site. math. �7�[RR'|Z��&���(�r�����O1���h�x���9�k��D(�hcL&�dN�e��%�]�8�gL����$� ���ݘ�\��;Kɭ�������. Z xn dx= xn+1 n+1 +C (n6= 1) 2. www.mathportal.org 5. ∫ = u +C u du ln 6. Integration Formulas PDF Download (Trig, Definite, Integrals, Properties) Integration Formulas PDF Download:- Hello friends, welcome to our website mynotesadda.com.Today our post is related to Maths topic, in this post we will provide you LInk to download all types of PDF … !��|�K���c���e�1}����Cv_L[���həs�wڎhV+�=�U��|�(ӖM���j�}D�A�g�S������R|�y��\VyK=��+�2���(K�)�:�"�(FZ=]�Ϸ���ԩ��d double integral gives us the volume under the surface z = f(x,y), just as a single integral gives the area under a curve. 1. As an Amazon associate we earn from qualifying purchases. Therefore, the desired function is f(x)=1 4 Techniques of Integration Over the next few sections we examine some techniques that are frequently successful when seeking antiderivatives of functions. To enable us to ﬁnd integrals of a wider range of functions than those normally given in a table of integrals we can make use of the following rules. dZk� l}�����Q%PK�4�c(��^�8�pm�t�CM�à�! (3x2 + 4)d dx {u} = 12 u.u d dx { 2 − 4x2 + 7x5} = 1 2 2 − 4x2 + 7x5 (−8x + 35x4) d dx {c} = 0 , c is a constant ddx {6} = 0 , since ≅ 3.14 is a constant. ©2005 BE Shapiro Page 3 This document may not be reproduced, posted or published without permission. x��\K���ﯘܸ�F��tJ�e�JQ�*��Ej�+��A��-�ߧ�yp���p%�t��$�t�� y�.��/g���/ξ{*�(�]\g�f�Ɍ��K�]\e���.Voo��K�b�����������x3�٭������������ǖ�ӳ!s�i���� �J�Ͼ���{aks������G4d!�F�c]Ȕ�5PȐ�1N2e�A��B4=��_Gf!�ѕ�'Zț� Kz)��� Z tanxdx= ln cosx +C 7. OpenStax is part of Rice University, which is a 501(c)(3) nonprofit. 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. Z secudu=ln|secu+tanu|+C 9. edu/ ~vhm/ Table. Partial Fractions 32 1.8. An elliptic integral is any integral of the general form f(x)= A(x)+B(x) C(x)+D(x)! 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. Z sec2 udu= tanu+C 13. If the power of the sine is odd and positive: Goal: ux cos i. There have been visitors to integral-table.com since 2004. sn+1 (11) tx … Table of Integrals for inverse trig forms, exponential/logarithmic forms, . %PDF-1.5 Gradshteyn and Ryzhik (GR) is the informal name of a comprehensive table of integrals originally compiled by the Russian mathematicians I. S. Gradshteyn and I. M. Ryzhik. 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. 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! Volumes 52 2.3. Home Research Teaching Students CV Software Google scholar. Second, we find a fast way to compute it. Z tanxdx= ln cosx +C 7. This page lists some of the most common antiderivatives then you must include on every digital page view the following attribution: Use the information below to generate a citation. Z cosecxdx= ln cosecx cotx +C 13. It's a pity this book is no longer in print, but it is still worth taking the trouble to buy a used copy. The integral tables are extensive and quite helpful; I agree that this is a must-have for physics majors. Z dx x = lnjxj+C 3. Z cscucotudu= − cscu+C 12. Table of Standard Integrals 1. Purchase Table of Integrals, Series, and Products - 8th Edition. The ﬁrst and second parts of the book, just as many sections, were written so that they could be read independently from each other. wolfram. Numerical Integration 41 1.10. Convert the remaining factors to cos( )x (using sin 1 cos22x x.) Z secxdx= ln secx+tanx +C 12. The copyright holder makes no representation about the accuracy, correctness, or Z xn dx= xn+1 n+1 +C (n6= 1) 2. 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. Save a du x dx sin( ) ii. 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. Integration using Tables and CAS 39 1.9. It's a pity this book is no longer in print, but it is still worth taking the trouble to buy a used copy. Print Book & E-Book. 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. html) View integral tables 5.pdf from MATH 2314 at University of Texas, Dallas. Trigonometric Integrals and Trigonometric Substitutions 26 1.7. Use the Print item from the "Question Help" menu to get a better print-out. Sometimes this is a simple problem, since it will be apparent that the function you wish to integrate is a derivative in some straightforward way. 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 Z cosxdx= sinx+C 6. View integral tables 5.pdf from MATH 2314 at University of Texas, Dallas. For example, they can help you get The integral tables are extensive and quite helpful; I agree that this is a must-have for physics majors. Trig Integrals: Integrals involving sin(x) and cos(x): Integrals involving sec(x) and tan(x): 1. (A) The Power Rule : Examples : d dx {un} = nu n−1. 1.5. Integration by Parts 21 1.6. sn+1 (11) tx … Gradshteyn and I.M. 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. Z Z n+1 x 1 1 eax xn dx = , n 6= −1 dx = ln |ax Unformatted text preview: Table of Integrals (page 1) ...Using the browser's print will lead to an undesirable print-out. integrals are given, as well as tables of Laplace, Mellin, and other transforms, which are used in the book. The OpenStax name, OpenStax logo, OpenStax book List of integrals of exponential functions 3 ( is the modified Bessel function of the first kind) References • Wolfram Mathematica Online Integrator (http:/ / integrals. Z ur du = ur+1 r +1 +C, r 6=−1 2. Not to mention their servers gave up the ghost turned into Zombies on 25 March 2015 (Brains! integrals are given, as well as tables of Laplace, Mellin, and other transforms, which are used in the book. This new edition contains many new results and corrections to the existing integrals. More about Areas 50 2.2. Table of Integrals for inverse trig forms, exponential/logarithmic forms, . The integral table in the frame above was produced TeX4ht for MathJax using the command sh ./makejax.sh integral-table the configuration file here, and the shell scripts ht5mjlatex and makejax.sh Use the Print item from the "Question Help" menu to get a better print-out. Table of Integrals . Z cscudu=ln|cscu−cotu|+C 10. Print Book & E-Book. Program Boise State University Boise, Idaho, USA gradywright@boisestate.edu . 1. â«undu=un+1n+1+C,nâ â1â«undu=un+1n+1+C,nâ â1, 5. â«sinudu=âcosu+Câ«sinudu=âcosu+C, 8. â«csc2udu=âcotu+Câ«csc2udu=âcotu+C, 9. â«secutanudu=secu+Câ«secutanudu=secu+C, 10. â«cscucotudu=âcscu+Câ«cscucotudu=âcscu+C, 11. â«tanudu=ln|secu|+Câ«tanudu=ln|secu|+C, 12. â«cotudu=ln|sinu|+Câ«cotudu=ln|sinu|+C, 13. â«secudu=ln|secu+tanu|+Câ«secudu=ln|secu+tanu|+C, 14. â«cscudu=ln|cscuâcotu|+Câ«cscudu=ln|cscuâcotu|+C, 15. â«dua2âu2=sinâ1ua+Câ«dua2âu2=sinâ1ua+C, 16. â«dua2+u2=1atanâ1ua+Câ«dua2+u2=1atanâ1ua+C, 17. â«duuu2âa2=1asecâ1ua+Câ«duuu2âa2=1asecâ1ua+C, 18. â«sin2udu=12uâ14sin2u+Câ«sin2udu=12uâ14sin2u+C, 19. â«cos2udu=12u+14sin2u+Câ«cos2udu=12u+14sin2u+C, 20. â«tan2udu=tanuâu+Câ«tan2udu=tanuâu+C, 21. â«cot2udu=âcotuâu+Câ«cot2udu=âcotuâu+C, 22. â«sin3udu=â13(2+sin2u)cosu+Câ«sin3udu=â13(2+sin2u)cosu+C, 23. â«cos3udu=13(2+cos2u)sinu+Câ«cos3udu=13(2+cos2u)sinu+C, 24. â«tan3udu=12tan2u+ln|cosu|+Câ«tan3udu=12tan2u+ln|cosu|+C, 25. â«cot3udu=â12cot2uâln|sinu|+Câ«cot3udu=â12cot2uâln|sinu|+C, 26. â«sec3udu=12secutanu+12ln|secu+tanu|+Câ«sec3udu=12secutanu+12ln|secu+tanu|+C, 27. â«csc3udu=â12cscucotu+12ln|cscuâcotu|+Câ«csc3udu=â12cscucotu+12ln|cscuâcotu|+C, 28. â«sinnudu=â1nsinnâ1ucosu+nâ1nâ«sinnâ2uduâ«sinnudu=â1nsinnâ1ucosu+nâ1nâ«sinnâ2udu, 29. â«cosnudu=1ncosnâ1usinu+nâ1nâ«cosnâ2uduâ«cosnudu=1ncosnâ1usinu+nâ1nâ«cosnâ2udu, 30. â«tannudu=1nâ1tannâ1uââ«tannâ2uduâ«tannudu=1nâ1tannâ1uââ«tannâ2udu, 31. â«cotnudu=â1nâ1cotnâ1uââ«cotnâ2uduâ«cotnudu=â1nâ1cotnâ1uââ«cotnâ2udu, 32. â«secnudu=1nâ1tanusecnâ2u+nâ2nâ1â«secnâ2uduâ«secnudu=1nâ1tanusecnâ2u+nâ2nâ1â«secnâ2udu, 33. â«cscnudu=â1nâ1cotucscnâ2u+nâ2nâ1â«cscnâ2uduâ«cscnudu=â1nâ1cotucscnâ2u+nâ2nâ1â«cscnâ2udu, 34. â«sinausinbudu=sin(aâb)u2(aâb)âsin(a+b)u2(a+b)+Câ«sinausinbudu=sin(aâb)u2(aâb)âsin(a+b)u2(a+b)+C, 35. â«cosaucosbudu=sin(aâb)u2(aâb)+sin(a+b)u2(a+b)+Câ«cosaucosbudu=sin(aâb)u2(aâb)+sin(a+b)u2(a+b)+C, 36. â«sinaucosbudu=âcos(aâb)u2(aâb)âcos(a+b)u2(a+b)+Câ«sinaucosbudu=âcos(aâb)u2(aâb)âcos(a+b)u2(a+b)+C, 37. â«usinudu=sinuâucosu+Câ«usinudu=sinuâucosu+C, 38. â«ucosudu=cosu+usinu+Câ«ucosudu=cosu+usinu+C, 39. â«unsinudu=âuncosu+nâ«unâ1cosuduâ«unsinudu=âuncosu+nâ«unâ1cosudu, 40. â«uncosudu=unsinuânâ«unâ1sinuduâ«uncosudu=unsinuânâ«unâ1sinudu, 41. â«sinnucosmudu=âsinnâ1ucosm+1un+m+nâ1n+mâ«sinnâ2ucosmudu=sinn+1ucosmâ1un+m+mâ1n+mâ«sinnucosmâ2uduâ«sinnucosmudu=âsinnâ1ucosm+1un+m+nâ1n+mâ«sinnâ2ucosmudu=sinn+1ucosmâ1un+m+mâ1n+mâ«sinnucosmâ2udu, 42. â«ueaudu=1a2(auâ1)eau+Câ«ueaudu=1a2(auâ1)eau+C, 43. â«uneaudu=1auneauânaâ«unâ1eauduâ«uneaudu=1auneauânaâ«unâ1eaudu, 44. â«eausinbudu=eaua2+b2(asinbuâbcosbu)+Câ«eausinbudu=eaua2+b2(asinbuâbcosbu)+C, 45. â«eaucosbudu=eaua2+b2(acosbu+bsinbu)+Câ«eaucosbudu=eaua2+b2(acosbu+bsinbu)+C, 46. â«lnudu=ulnuâu+Câ«lnudu=ulnuâu+C, 47. â«unlnudu=un+1(n+1)2[(n+1)lnuâ1]+Câ«unlnudu=un+1(n+1)2[(n+1)lnuâ1]+C, 48. â«1ulnudu=ln|lnu|+Câ«1ulnudu=ln|lnu|+C, 51. â«tanhudu=lncoshu+Câ«tanhudu=lncoshu+C, 52. â«cothudu=ln|sinhu|+Câ«cothudu=ln|sinhu|+C, 53. â«sechudu=tanâ1|sinhu|+Câ«sechudu=tanâ1|sinhu|+C, 54. â«cschudu=ln|tanh12u|+Câ«cschudu=ln|tanh12u|+C, 55. â«sech2udu=tanhu+Câ«sech2udu=tanhu+C, 56. â«csch2udu=âcothu+Câ«csch2udu=âcothu+C, 57. â«sechutanhudu=âsechu+Câ«sechutanhudu=âsechu+C, 58. â«cschucothudu=âcschu+Câ«cschucothudu=âcschu+C, 59. â«sinâ1udu=usinâ1u+1âu2+Câ«sinâ1udu=usinâ1u+1âu2+C, 60. â«cosâ1udu=ucosâ1uâ1âu2+Câ«cosâ1udu=ucosâ1uâ1âu2+C, 61. â«tanâ1udu=utanâ1uâ12ln(1+u2)+Câ«tanâ1udu=utanâ1uâ12ln(1+u2)+C, 62. â«usinâ1udu=2u2â14sinâ1u+u1âu24+Câ«usinâ1udu=2u2â14sinâ1u+u1âu24+C, 63. â«ucosâ1udu=2u2â14cosâ1uâu1âu24+Câ«ucosâ1udu=2u2â14cosâ1uâu1âu24+C, 64. â«utanâ1udu=u2+12tanâ1uâu2+Câ«utanâ1udu=u2+12tanâ1uâu2+C, 65. â«unsinâ1udu=1n+1[un+1sinâ1uââ«un+1du1âu2],nâ â1â«unsinâ1udu=1n+1[un+1sinâ1uââ«un+1du1âu2],nâ â1, 66. â«uncosâ1udu=1n+1[un+1cosâ1u+â«un+1du1âu2],nâ â1â«uncosâ1udu=1n+1[un+1cosâ1u+â«un+1du1âu2],nâ â1, 67. â«untanâ1udu=1n+1[un+1tanâ1uââ«un+1du1+u2],nâ â1â«untanâ1udu=1n+1[un+1tanâ1uââ«un+1du1+u2],nâ â1, 68. â«a2+u2du=u2a2+u2+a22ln(u+a2+u2)+Câ«a2+u2du=u2a2+u2+a22ln(u+a2+u2)+C, 69. â«u2a2+u2du=u8(a2+2u2)a2+u2âa48ln(u+a2+u2)+Câ«u2a2+u2du=u8(a2+2u2)a2+u2âa48ln(u+a2+u2)+C, 70. â«a2+u2udu=a2+u2âaln|a+a2+u2u|+Câ«a2+u2udu=a2+u2âaln|a+a2+u2u|+C, 71. â«a2+u2u2du=âa2+u2u+ln(u+a2+u2)+Câ«a2+u2u2du=âa2+u2u+ln(u+a2+u2)+C, 72. â«dua2+u2=ln(u+a2+u2)+Câ«dua2+u2=ln(u+a2+u2)+C, 73. â«u2dua2+u2=u2(a2+u2)âa22ln(u+a2+u2)+Câ«u2dua2+u2=u2(a2+u2)âa22ln(u+a2+u2)+C, 74. â«duua2+u2=â1aln|a2+u2+au|+Câ«duua2+u2=â1aln|a2+u2+au|+C, 75. â«duu2a2+u2=âa2+u2a2u+Câ«duu2a2+u2=âa2+u2a2u+C, 76. â«du(a2+u2)3/2=ua2a2+u2+Câ«du(a2+u2)3/2=ua2a2+u2+C, 77. â«u2âa2du=u2u2âa2âa22ln|u+u2âa2|+Câ«u2âa2du=u2u2âa2âa22ln|u+u2âa2|+C, 78. â«u2u2âa2du=u8(2u2âa2)u2âa2âa48ln|u+u2âa2|+Câ«u2u2âa2du=u8(2u2âa2)u2âa2âa48ln|u+u2âa2|+C, 79. â«u2âa2udu=u2âa2âacosâ1a|u|+Câ«u2âa2udu=u2âa2âacosâ1a|u|+C, 80. â«u2âa2u2du=âu2âa2u+ln|u+u2âa2|+Câ«u2âa2u2du=âu2âa2u+ln|u+u2âa2|+C, 81. â«duu2âa2=ln|u+u2âa2|+Câ«duu2âa2=ln|u+u2âa2|+C, 82. â«u2duu2âa2=u2u2âa2+a22ln|u+u2âa2|+Câ«u2duu2âa2=u2u2âa2+a22ln|u+u2âa2|+C, 83. â«duu2u2âa2=u2âa2a2u+Câ«duu2u2âa2=u2âa2a2u+C, 84. â«du(u2âa2)3/2=âua2u2âa2+Câ«du(u2âa2)3/2=âua2u2âa2+C, 85. â«a2âu2du=u2a2âu2+a22sinâ1ua+Câ«a2âu2du=u2a2âu2+a22sinâ1ua+C, 86. â«u2a2âu2du=u8(2u2âa2)a2âu2+a48sinâ1ua+Câ«u2a2âu2du=u8(2u2âa2)a2âu2+a48sinâ1ua+C, 87. â«a2âu2udu=a2âu2âaln|a+a2âu2u|+Câ«a2âu2udu=a2âu2âaln|a+a2âu2u|+C, 88. â«a2âu2u2du=â1ua2âu2âsinâ1ua+Câ«a2âu2u2du=â1ua2âu2âsinâ1ua+C, 89. â«u2dua2âu2=âu2a2âu2+a22sinâ1ua+Câ«u2dua2âu2=âu2a2âu2+a22sinâ1ua+C, 90. â«duua2âu2=â1aln|a+a2âu2u|+Câ«duua2âu2=â1aln|a+a2âu2u|+C, 91. â«duu2a2âu2=â1a2ua2âu2+Câ«duu2a2âu2=â1a2ua2âu2+C, 92. â«(a2âu2)3/2du=âu8(2u2â5a2)a2âu2+3a48sinâ1ua+Câ«(a2âu2)3/2du=âu8(2u2â5a2)a2âu2+3a48sinâ1ua+C, 93. â«du(a2âu2)3/2=ua2a2âu2+Câ«du(a2âu2)3/2=ua2a2âu2+C, 94. â«2auâu2du=uâa22auâu2+a22cosâ1(aâua)+Câ«2auâu2du=uâa22auâu2+a22cosâ1(aâua)+C, 95. â«du2auâu2=cosâ1(aâua)+Câ«du2auâu2=cosâ1(aâua)+C, 96. â«u2auâu2du=2u2âauâ3a262auâu2+a32cosâ1(aâua)+Câ«u2auâu2du=2u2âauâ3a262auâu2+a32cosâ1(aâua)+C, 97. â«duu2auâu2=â2auâu2au+Câ«duu2auâu2=â2auâu2au+C, 98. â«udua+bu=1b2(a+buâaln|a+bu|)+Câ«udua+bu=1b2(a+buâaln|a+bu|)+C, 99. â«u2dua+bu=12b3[(a+bu)2â4a(a+bu)+2a2ln|a+bu|]+Câ«u2dua+bu=12b3[(a+bu)2â4a(a+bu)+2a2ln|a+bu|]+C, 100. â«duu(a+bu)=1aln|ua+bu|+Câ«duu(a+bu)=1aln|ua+bu|+C, 101. â«duu2(a+bu)=â1au+ba2ln|a+buu|+Câ«duu2(a+bu)=â1au+ba2ln|a+buu|+C, 102. â«udu(a+bu)2=ab2(a+bu)+1b2ln|a+bu|+Câ«udu(a+bu)2=ab2(a+bu)+1b2ln|a+bu|+C, 103. â«uduu(a+bu)2=1a(a+bu)â1a2ln|a+buu|+Câ«uduu(a+bu)2=1a(a+bu)â1a2ln|a+buu|+C, 104. â«u2du(a+bu)2=1b3(a+buâa2a+buâ2aln|a+bu|)+Câ«u2du(a+bu)2=1b3(a+buâa2a+buâ2aln|a+bu|)+C, 105. â«ua+budu=215b2(3buâ2a)(a+bu)3/2+Câ«ua+budu=215b2(3buâ2a)(a+bu)3/2+C, 106. â«udua+bu=23b2(buâ2a)a+bu+Câ«udua+bu=23b2(buâ2a)a+bu+C, 107. â«u2dua+bu=215b3(8a2+3b2u2â4abu)a+bu+Câ«u2dua+bu=215b3(8a2+3b2u2â4abu)a+bu+C, 108. â«duua+bu=1aln|a+buâaa+bu+a|+C,ifa>0=2âatanâ1a+buâa+C,ifa<0â«duua+bu=1aln|a+buâaa+bu+a|+C,ifa>0=2âatanâ1a+buâa+C,ifa<0, 109. â«a+buudu=2a+bu+aâ«duua+buâ«a+buudu=2a+bu+aâ«duua+bu, 110. â«a+buu2du=âa+buu+b2â«duua+buâ«a+buu2du=âa+buu+b2â«duua+bu, 111. â«una+budu=2b(2n+3)[un(a+bu)3/2ânaâ«unâ1a+budu]â«una+budu=2b(2n+3)[un(a+bu)3/2ânaâ«unâ1a+budu], 112. â«undua+bu=2una+bub(2n+1)â2nab(2n+1)â«unâ1dua+buâ«undua+bu=2una+bub(2n+1)â2nab(2n+1)â«unâ1dua+bu, 113. â«duuna+bu=âa+bua(nâ1)unâ1âb(2nâ3)2a(nâ1)â«duunâ1a+buâ«duuna+bu=âa+bua(nâ1)unâ1âb(2nâ3)2a(nâ1)â«duunâ1a+bu. 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