Second, we find a fast way to compute it. 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 Integration by Parts 21 1.6. Section 7-5 : Proof of Various Integral Properties. Gradshteyn and I.M. This book is Creative Commons Attribution-NonCommercial-ShareAlike License First, a double integral is defined as the limit of sums. Integration by Parts 21 1.6. Z cosxdx= sinx+C 6. View Calc 2 TABLE OF INTEGRATION FORMULASwithlines.pdf from MATH 22 at Howard University. Table of Integrals, Series, and Products Seventh Edition I.S. Z xn dx= xn+1 n+1 +C (n6= 1) 2. Gradshteyn and I.M. Z ur du = ur+1 r +1 +C, r 6=−1 2. Home Research Teaching Students CV Software Google scholar. More about Areas 50 2.2. 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. Applications of Integration 50 2.1. Z e xdx= e +C 4. More about Areas 50 2.2. 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! The integral tables are extensive and quite helpful; I agree that this is a must-have for physics majors. View integral tables.pdf from MATH 220 at University of Missouri, Kansas City. 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 9780123849335, 9780123849342 Table of Integrals . Unformatted text preview: Table of Integrals (page 2) ...Using the browser's print will lead to an undesirable print-out. This page lists some of the most common antiderivatives Try this for yourself. Partial Fractions 32 1.8. Second, we find a fast way to compute it. We recommend using a 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. 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 The integral tables are extensive and quite helpful; I agree that this is a must-have for physics majors. Therefore, the desired function is f(x)=1 4 Table of Integrals∗ Basic Forms Integrals with Logarithms √ x ax + bdx = Z Z 1 x dx = xn+1 n+1 Z 1 dx = ln 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. >> 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. Use the Print item from the "Question Help" menu to get a better print-out. Use the Print item from the "Question Help" menu to get a better print-out. Improper Integrals 46 Chapter 2. Print Book & E-Book. View Table of integrals.pdf from MA 3253 at Mississippi State University. pdf doc ; Integration Techniques - A collection of problems using various integration techniques. Z sinudu= − cosu+C 5. We also Cover Basic Topics like Maths, Geography, H istory, Polity, etc and study materials including previous Year … 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.. Z secxdx= ln secx+tanx +C 12. CHAPTER 14 Multiple Integrals 14.1 Double Integrals 4 This chapter shows how to integrate functions of two or more variables. STANDARD INTEGRALS are provided. jsp) • V. H. Moll, The Integrals in Gradshteyn and Ryzhik (http:/ / www. 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 %PDF-1.5 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 Z sec2 udu= tanu+C 13. Z Z Z Z Z Z Z Z Z Table of integrals Note: Integrating constant is omitted. integrals are given, as well as tables of Laplace, Mellin, and other transforms, which are used in the book. Use the Print item from the "Question Help" menu to get a better print-out. The key idea is to replace a double integral by two ordinary "single" integrals. For example, faced with Z x10 dx If the power of the sine is odd and positive: Goal: ux cos i. Save a du x dx sin( ) ii. Home Research Teaching Students CV Software Google scholar. Table of Integrals . tulane. Table of Integrals 1. About me. Z 1 u du =ln|u|+C 3. 2. ∫ = u +C u du ln 6. Not to mention their servers gave up the ghost turned into Zombies on 25 March 2015 (Brains! Program Boise State University Boise, Idaho, USA [email protected] . First, a double integral is defined as the limit of sums. Convert the remaining factors to cos( )x (using sin 1 cos22x x.) Table of Integrals, Series, and Products Seventh Edition I.S. List of integrals of exponential functions 3 ( is the modified Bessel function of the first kind) References • Wolfram Mathematica Online Integrator (http:/ / integrals. Professor Department of Mathematics Computing Ph.D. !��|�K���c���e�1}����Cv_L[���həs�wڎhV+�=�U��|�(ӖM���j�}D�A�g�S������R|�y��\VyK=��+�2���(K�)�:�"�(FZ=]�Ϸ���ԩ��d Purchase Table of Integrals, Series, and Products - 7th Edition. Z e xdx= e +C 4. Print Book & E-Book. 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. 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. double integral gives us the volume under the surface z = f(x,y), just as a single integral gives the area under a curve. math. #ӻ�D�"$Ӧ�W��6dKa��e�π;N�2i����~�8�ϙ�.� ��AF1�"��;{��gW��ˌ3$|C�h����f����@xC $The OpenStax name, OpenStax logo, OpenStax book Techniques of Integration Over the next few sections we examine some techniques that are frequently successful when seeking antiderivatives of functions. 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 … 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. Z dx x = lnjxj+C 3. The integral of kkff(x) where k is a constant A constant factor in an integral can be moved outside the integral sign … Partial Fractions 32 1.8. Improper Integrals 46 Chapter 2. 1. Table of Integrals∗ Basic Forms Integrals with Logarithms √ x ax + bdx = Z Z 1 x dx = xn+1 n+1 Z 1 dx = ln The ﬁrst and second parts of the book, just as many sections, were written so that they could be read independently from each other. Program Boise State University Boise, Idaho, USA [email protected] . STANDARD INTEGRALS are provided. Volumes 52 2.3. dZk� l}�����Q%PK�4�c(��^�8�pm�t�CM�à�! 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. Its full title today is Table of Integrals, Series, and Products.. A: TABLE OF BASIC DERIVATIVES Let u = u(x) be a differentiable function of the independent variable x, that is u(x) exists. Table of Integrals for inverse trig forms, exponential/logarithmic forms, . ©2005 BE Shapiro Page 3 This document may not be reproduced, posted or published without permission. citation tool such as, Authors: Gilbert Strang, Edwin âJedâ Herman. /Length 3274 The copyright holder makes no representation about the accuracy, correctness, or This new edition contains many new results and corrections to the existing integrals. Z sinxdx= cosx+C 5. Z Z n+1 x 1 1 eax xn dx = , n 6= −1 dx = ln |ax Professor Department of Mathematics Computing Ph.D. 7 0 obj << Z tanxdx= ln cosx +C 7. Z dx x = lnjxj+C 3. ∫e du =eu +C Example 2: Evaluate ∫( ) 4x2 −5x3 +12 dx To evaluate this problem, use the first four Integral Formulas. Trigonometric Integrals and Trigonometric Substitutions 26 1.7. 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, Except where otherwise noted, textbooks on this site Arc Length, Parametric Curves 57 2.4. wolfram. Z secudu=ln|secu+tanu|+C 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. 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. Trig Integrals: Integrals involving sin(x) and cos(x): Integrals involving sec(x) and tan(x): 1. The ﬁrst and second parts of the book, just as many sections, were written so that they could be read independently from each other. not be reproduced without the prior and express written consent of Rice University. stream 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… 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. Z sinxdx= cosx+C 5. html) ISBN 9780123736376, 9780080471112 Section 7-5 : Proof of Various Integral Properties. For other types of functions, see lists of integrals Arc Length, Parametric Curves 57 2.4. This allows the reader to quickly get to the heart of the matter. An elliptic integral is any integral of the general form f(x)= A(x)+B(x) C(x)+D(x)! Z cscudu=ln|cscu−cotu|+C 10. View integral tables 5.pdf from MATH 2314 at University of Texas, Dallas. Print Book & E-Book. 4.0 and you must attribute OpenStax. Z secutanudu= secu+C 11. Unformatted text preview: Table of Integrals (page 1) ...Using the browser's print will lead to an undesirable print-out. ∫e du =eu +C Example 2: Evaluate ∫( ) 4x2 −5x3 +12 dx To evaluate this problem, use the first four Integral Formulas. Want to cite, share, or modify this book? 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. Table of Standard Integrals 1. 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. 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. Z tanxdx= ln cosx +C 7. 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 cscucotudu= − cscu+C 12. 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 For example, they can help you get Z cosecxdx= ln cosecx cotx +C 13. 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. Creative Commons Attribution-NonCommercial-ShareAlike License 4.0 license. Z secxdx= ln secx+tanx +C 12. 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. Integration using Tables and CAS 39 1.9. Save a du x dx sin( ) ii. Trigonometric Integrals and Trigonometric Substitutions 26 1.7. then you must include on every digital page view the following attribution: Use the information below to generate a citation. Table of Integrals for inverse trig forms, exponential/logarithmic forms, . Purchase Table of Integrals, Series, and Products - 8th Edition. sn+1 (11) tx … There have been visitors to integral-table.com since 2004. 1. 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. edu/ ~vhm/ Table. Numerical Integration 41 1.10. 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 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.. Â© 1999-2020, Rice University. Elliptic integrals can be viewed as generalizations of the inverse trigonometric functions. If the power of the sine is odd and positive: Goal: ux cos i. Z cosec2 xdx= cotx+C 11. ® and AP ® are trademarks registered and/or owned by the College Board, which is not affiliated with, and does not endorse, this site. Z xn dx= xn+1 n+1 +C (n6= 1) 2. 1.5. 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 Unformatted text preview: Table of Integrals (page 1) ...Using the browser's print will lead to an undesirable print-out. This new edition contains many new results and corrections to the existing integrals. About me. The result of ﬁnding an indeﬁnite integral is usually a function plus a constant of integration. 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. 1 of Z cotudu=ln|sinu|+C 8. Z cosudu= sinu+C 6. 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. Purchase Table of Integrals, Series, and Products - 7th Edition. Z Z n+1 x 1 1 eax xn dx = , n 6= −1 dx = ln |ax Z cosxdx= sinx+C 6. 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. "�0+\�� {Z'�˰�������-�������Ch�X��p aN���� �pr"��ʿ�)����C�i�٣&g[e=&���O%�D��:�V��+^�� �qH9&���*�[email protected]��2mU�g:��8�fng��W�����K��|�%Qi���aHmS��VU�� u ddx {(x3 + 4x + 1)3/4} = 34 (x3 + 4x + 1)−1/4. ISBN 9780123736376, 9780080471112 Z cotxdx= ln sinx +C 8. It's a pity this book is no longer in print, but it is still worth taking the trouble to buy a used copy. ������i�� �Q���}R|tL���3�� pz��Q洩z�*���z��>ί Volumes 52 2.3. ∫ = u +C u du ln 6. Z Z Z Z Z Z Z Z Z Table of integrals Note: Integrating constant is omitted. 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. Trig Integrals: Integrals involving sin(x) and cos(x): Integrals involving sec(x) and tan(x): 1. Our mission is to improve educational access and learning for everyone. Integration using Tables and CAS 39 1.9. This allows the reader to quickly get to the heart of the matter. Z tanudu=ln|secu|+C 7. Integrals of Trig. Z cotxdx= ln sinx +C 8. Â© Sep 2, 2020 OpenStax. Use the Print item from the "Question Help" menu to get a better print-out. sn+1 (11) tx … 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. (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. www.mathportal.org 5. 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. com/ index. 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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