毕业论文-有理函数不定积分的研究.doc

本 科 毕 业 论 文 题 目： 有理函数不定积分的研究 The Study of Indefinite Integral of Rational Function 姓 名 学 院 贵 州 民 族 大 学 专业、年级 数学与应用数学 2013 级 二一五 年 3 月 摘 要 随着数学领域的不断创新，有理函数不定积分不仅是微积分学中的一个重要内容，也是不定积分教学中的一个重点和难点。我们在学习不定积分的基础上来对有理函数不定积分做进一步研究，探讨有理函数的可积性。对数学分析中一些比较复杂问题学会初步处理，对于简单的有理函数不定积分的问题 ，我们可以用观察法、配项法进行拆分，而比较复杂的情况，就要用到凑微分法、待定系数法进行求解。我们将重点在于对有理函数积分方法做一个系统的全面归纳总结，这在我们今后的工作和研究中起着重要的作用。关键词：不定积分、有理函数、凑微分法、配项法、待定系数法ABSTRCT With the continuous innovation in the field of mathematics, the indefinite integral of rational function not only becomes an important content of calculus, but also the focus and difficulty of the indefinite integral teaching. We learn indefinite integral on the basis of further study about the indefinite integral of rational function for exploring the integrability of a rational function. Learning to deal with some complex problems in the mathematical analysis .For some simple indefinite integral of rational function problems, we can use the method of observation, a method to split, while for more complex situations, gather together differential and method of undetermined coefficients would be took into account. We will focus on making a systematic conclusion of rational function integral method , which plays an important role in our future study.Key words: differential, indefinite integral, the rational function, gather together with a method, the method of undetermined coefficients目 录1 引言12 不定积分的概念、定义及不定积分的几个积分方法32.1原函数与不定积分的概念理解32.2积分的运算性质及积分方法52.2.1直接积分法52.2.2 凑微分法62.2.3分部积分法73 有理函数的不定积分83.1 有理函数的一般形式及分类83.2 真假有理函数的转化83.3 有理函数不定积分的积分方法93.1.1 用凑微分法求有理函数不定积分93.1.2 用配项法求有理函数不定积分94 有理函数的运用与推广104.1 有限个函数推广到n个函数代数和114.2 多项式分解125 毕业论文总结14参考文献15致 谢16 1 引言 通过对有理函数不定积分相关问题的分析，我们能够了解到有理函数不定积分总是能“积”出来，即有理函数不定积分总能够用初等函数：有理函数、对数函数、反正切函数表示出来。一般来说，在数学领域里，一种运算的出现都伴随着它的逆运算，例如，有加就有减，有乘就有除，有乘方就有开方，等等。导数运算也不例外，我们通过对有理函数不定积分相关内容学习探讨，就能解决很多数学领域的问题。一般来说，求不定积分要比求导数困难得多。这是因为导数的定义是构造性的，如果函数存在导数，根据导数运算法则和导数公式或者导数定义，按求导运算程序，总能够求出函数的导数。但是，求函数的不定积分则不然，根据不定积分运算法则和不定积分公式只能求出很少一部分比较简单的函数的不定积分，而对更多函数的不定积分要因函数不同的形式或不同类型选用不同的方法。因此，求不定积分有很大的灵活性。有理函数存在初等函数的原函数（不定积分），这是有理函数集合一个理想的性质。如果求一个函数的不定积分，只要选择适当的换元，将被积函数化为有理函数，那么这个不定积分总是能够“积”出来的。有理函数不定积分虽然是微积分学里面的一个分支，但是却担当了相当重要的角色，它渗透到了微积分学的很多领域，成为很重要的工具。随着数学领域的不断创新，有理函数不定积分不仅是微积分学中的一个重要内容，也是不定积分教学中的一个重点和难点。我们在学习不定积分的基础上来对有理函数不定积分做进一步的研究，探讨有理函数的可积性。对数学分析中一些比较复杂问题学会初步处理，对于简单的有理函数不定积分的问题，我们可以用凑微分法、待定系数法进行求解。我们将重点在于对有理函数积分方法做一个系统的全面归纳总结，这在我们今后的工作和研究中起着重要的作用。微积分学涉及很多内容，同时微积分学是我们研究数学的一个有力工具，它使各个章节的内容联系更加紧密，有助于学生对数学的深入学习。微积分的思想方法是数学发展学生智力的关键所在，对研究有理函数不定积分有重要作用。我们通过分析探讨它的概念、解题方法、地位作用等等对我们今后在研究微积分学上有着非常重要的作用和意义。2 不定积分的概念、定义及不定积分的几个积分方法 2.1原函数与不定积分的概念理解 原函数的定义：设函数在区间有定义，存在函数，若，有，则称函数是在区间的原函数或简称是的原函数。 例如：,=,即是的原函数。，,，即也是的原函数。 由例子可知，若函数存在原函数，则这个原函数加上任意常数,即也是函数的原函数，于是，一个函数存在原函数，那么它必有无限多个原函数。 定理1、若是函数在区间的一个原函数，则函数f(x)的无限多个原函数仅限于（）的形式。探讨研究：已知是函数的一个原函数，有，。设是函数的任意（注意“任意”二字）一个原函数，即，有，两式相减，有以下的式子：或。 根据以前的推论可知，（是某个常数）或，即函数的任意一个原函数）都是的形式。 这个定理指出，一个函数的无限多个原函数彼此仅相差一个常数。如果欲求函数的所有的原函数，只需求出函数f(x)只需数的一个原函求出函数，然后再加上任意常数，就得到了函数的所有的原函数。 不定积分的定义：函数f(x)在区间I的所有原函数（）称为函数f(x)的不定积分，表为：（），其中f(x)称为被积函数， 称为被积表达式，称为积分常数。 注意：根据原函数的定义，当我们说是的原函数，总是对一个特定区间而言的，则对任意常数，有也是上的原函数，这里的常数也是对区间而言的，也就是说，对同一个区间，必须是同一个数。例如：函数 在x=0点不可导，我们说，在，,是的原函数；在是 的原函数；但不能说在上，是的原函数。由此可见，一个函数的不定积分既不是一个数，也不是一个函数，而是一个函数族。2.2积分的运算性质及积分方法2.2.1直接积分法 直接积分法就是利用积分公式和积分的基本性质求不定积分的方法，并且直接积分法关键就是对被积函数不管使用什么样的变形，最后都要利用不定积分的运算性质和基本的积分公式，直接求出不定积分。因此，对被积函数的变形要遵循“有的放矢”的原则，进而为选择适当的基本积分公式打开方便之门。 性质一、设函数在区间上存在原函数，,则函数也是在上存在原函数，且 。 性质二、设函数都在区间上存在原函数，则函数或也在上存在原函数，且 例：求 解： = = =2.2.2 凑微分法 凑微分法的基本思想是：把所求的被积函数通过适当的变量代换后，化成积分公式中的某一被积形式，然后代积分公式求出积分结果。该积分的关键是：将被积表达式凑成两部分，一部分是复合函数，其中外函数是基本积分公式中的某一被积形式，另一部分是内函数的微分，在计算的同时要注意凑微分的过程中系数的调整。有一些不定积分，将积分变量进行一定的变换后就能由基本公式求出所需的积分。例如求 这个不定分与公式不同，区别在于后者的被积函数是积分变量x的直接函数，而前者的被积函数是积分变量x的复合函数，故不能直接应用公式，但这个被积函数是与之积，且，这样所求积分可以改写成，再令，则就与基本积分公式相同，然后再代回原来的变量，就求得不定积分: = = =例：求 解：令,则du=2dx, 将和代入被积表达式，得： =2.2.3分部积分法 在积分时，还会遇到这类积分，如，等积分，它们的被积函数都是两个不同类型的函数之积，是无法用换元 积分法积出的，对这类积分可采用分部积分法来解决。 分部积分法，可以利用两个函数乘积的求导公式推导出来： 设函数都有连续的导数，则由函数乘积的导数公式, 移项得，对等式两边作不定积分得： 即 ，这就是分部积分公式。当积分不易计算，而积分比较容易计算时，就可以利用这个公式。此方法的关键是适当地选取和，使等式右边的积分变得容易计算些。 例：求 解：设由分部积分公式，有： = = =3 有理函数的不定积分 定义：由两个多项式函数的商所表示的函数称为有理函数。3.1 有理函数的一般形式及分类 一般形式：，其中与都是多项式。 分类：若的次数大于或等于的次数，称为有理假分式，若的次数小于的次数，称为有理真分式。 其中，（1），真分式 （2），假分式3.2 真假有理函数的转化 （1）任意有理假分式，用除，总能化为多项式与有理真分式之和，即，其中的次数低于的次数。 例：因为多项式的不定积分易求，所以求有理函数的不定积分关键在于求有理真分式的不定积分。3.3 有理函数不定积分的积分方法 对于有理函数的不定积分有几种方法，用凑微分法求有理 数不定积分，用配项法求有理函数不定积分，用待定系数法求有理函数不定积分。3.1.1 用凑微分法求有理函数不定积分 定理：若函数在可导，且，有，则函数存在原函数，即： = 例：求解：= 该解法用了添项与除以5的技巧凑微分，添（减）项与同乘（除）是积分运算中重要的变形技巧，需要仔细领会。3.1.2 用配项法求有理函数不定积分 有些有理函数可以通过对分子进行配项，然后将整个分式分成两项或n项之和，再针对每项利用积分公式积分即可。 例：求 解：设有 解得：A=4，B=-1，C=2即 = =该方法是将分子分母逐一分项，由一个复杂的整式化为几个简单的整式，然后对应系数相等而求解，最后积分达到解题的目的，也是有理函数不定积分常用的一种方法。 4 有理函数的运用与推广 4.1 有限个函数推广到n个函数代数和 法则： 这个法则可推广到n个（有限个）函数，即n个函数代数和的不定积分等于n个函数不定积分的代数和。 于是，在允许相差一个任意常数的意义下，不定积分这个运算恰好是求导运算的逆运算。因为积分运算是导数运算的逆运算，所以导数公式中的每一公式反过来就得到了不定积分的公式。 不定积分公式表与导数公式表不同，导数公式表全是基本初等函数的导数公式表，而不定积分公式的原函数不都是基本初等函数，例如：原函数是 ， ，等等都不是基本初等函数。 求函数不定积分最后都要归结为不定积分表所列的初等函数的不定 积分。应用不定积分法则和不定积分公式能够求一些简单函数的不定积分： 例 求 解 = =4 = = 注意：等式右端的每一个不定积分在区间都有一个任意常数，因为有限个任意常数的代数和还是一个任意常数，所以上式只写一个任意常数即可。 例2 求 解 = =4.2 多项式分解 任何实系数多项式总可以分解为实系数一次或二次因式的乘积：（部分分式展开定理）因此有理函数的积分问题就归结为求 和 。例 将化为部分分式，并计算解：故例 求 解：根据分解式，计算得 例 求 解：例 求 解： 微积分学涉及很多内容，同时微积分学是我们研究数学的一个有力工具，它使各个章节的内容联系更加紧密，有助于学生对数学的深入学习。微积分的思想方法是数学发展学生智力的关键所在，对研究有理函数不定积分有重要作用。5 毕业论文总结 在学校安排即将毕业的学生要写毕业论文后，我开始了我的毕业论文工作，时至今日，论文基本完成。从最初的茫然，到慢慢的进入状态，再到对思路逐渐的清晰，整个写作过程难以用语言来表达。历经了长时间的奋战，紧张而又充实的毕业设计终于落下了帷幕，我拥有了无数难忘的回忆和收获。通过查阅资料和所学专业知识，我对有理函数的不定积分的研究得到以下结论：解决有理函数不定积分的主要方法大致有分部积分法、凑微分法、直接积分法几类，直接积分法是利用积分公式和积分的基本性质求不定积分，并且直接积分法关键就是对被积函数不管使用什么样的变形，最后都要利用不定积分的运算性质和基本的积分公式，直接求出不定积分；凑微分法的思想是把所求的被积函数通过适当的变量代换后，化成积分公式中的某一被积形式，然后代积分公式求出积分结果。该积分的关键是将被积表达式凑成两部分，一部分是复合函数，其中外函数是基本积分公式中的某一被积形式，另一部分是内函数的微分，在计算的同时要注意凑微分的过程中系数的调整；有理函数不定积分作为微积分学的一个分支占有很重要的地位，在这个运算过程中具有很大的灵活性，但是有理函数不定积分总能积出来，即有理函数的不定积分总能用初等函数：有理函数、对数函数和反正切函数表示出来。在这段时间里，我与老师建立了浓厚的师生情谊。老师的谆谆教导，使我体会了学习的乐趣。我与身边许多同学，也建立了良好的学习关系，互帮互助，克服难关。现在我已经接近毕业，锻炼了自我的动手和分析问题能力，受益匪浅。 在这次毕业论文撰写中也使我们的同学关系更进一步了，同学之间互相帮助，有什么不懂的大家在一起商量，听听不同的看法对我们更好的理解知识，所以在这里非常感谢帮助我的同学。 我的导师和专业老师，是你们的细心指导和关怀，使我能够顺利的完成毕业论文。在我的学业和论文的研究工作中无不倾注着老师们辛勤的汗水和心血，老师的严谨治学态度、渊博的知识、无私的奉献精神使我深受启迪。在此，我表示衷心的感谢和深深的敬意。通过本次论文写作，使我掌握了微积分领域的一些相关有理函数不定积分的重要知识。 参考文献朱来义.高等学校经济管理学科数学基础微积分高等教育出版社.2004 东北师范大学.数学分析上册. 人民教育出版社.1983刘玉莲 林钉等编.数学分析第五版 .高等教育出版社.2009纪乐刚.数学分析.华东师范大学出版社.1989朱学炎.微积分大学数理化适用手册.上海科学技术社.1989董洗印 杨静懿.微积分.对外经济贸易大学出版社.1997 致 谢 本论文是在其他老师的亲切关怀和悉心指导下完成的，他严肃的工作态度和对人、对事高度的责任感以及持之以恒的不懈帮助，深深地感染和激励着我。李老师不仅在学业上给我以精心指导，而且在思想、生活上教育我如何为人处事，更是在工作上教育我要有个人严谨的作风，这都对我论文的完成有难以预计的帮助。在此我谨向李老师致以诚挚的谢意和崇高的敬意。另外我还要感谢与我一起快乐成长的毕业论文小组同学们，正是由于你们的帮助和支持，我才能克服一个一个的困难和疑惑，直至本文的顺利完成。在论文即将完成之际，我的心情无比轻松和畅快，从开始进入课题到论文的顺利完成，有多少可敬的师长、同学、朋友给了我无言的帮助，在这里请接受我诚挚的谢意!此外我还要感谢培养我长大含辛茹苦的父母，谢谢你们！最后，再次对关心、帮助我的老师和同学表示衷心地感谢！a您好，为你提供优秀的毕业论文参考资料，请您删除以下内容，O(_)O谢谢！AnationalsurveywasrecentlylaunchedtoevaluatetheeyehealthofChinesechildrenandteenagers.OnJune6,ChinasannualNationalDayforEyeCare,theChinaYouthDevelopmentServiceCenterandZhejiangMedicine,aleadinglistedChinesepharmaceuticalcompany,jointlyannouncedthekickoffofthesurvey.Inaboutonemonth,aquestionnairecompiledbytopeyecaremedicalexpertsinChinawillbedistributedthroughmultipleonlinepartners,includingH,aswellasthroughofflinesurveyeventsheldinuniversities,middleschoolsandprimaryschoolsacrossthecountry.Areportwillbereleasedbasedonthesurveystatisticsandanalysis,andmostimportantly,guidelinesforparentsandyouthonhowtocarefortheeyesandpreventmyopiaagrowingprobleminChinasdigitizedsociety,willalsobeattachedonthereport.Myopiaisnotonlyadiseasethatmakespeopleseethingsblurrily,butalsoleadstoseverecomplications,suchasglaucoma(increasedpressurewithintheeyeball),andcancauseblindness,saidZhouYuehua,anestablishedeyecarespecialistwithBeijingTongrenHospital.Itisveryimportantforparentsandchildrentoknowabouttheriskandcarefortheireyes.Thereareabout450millionmyopiapatientsinChina.AmongChinesemyopiapatients,30millionareseverepatients,accordingtoZhouTheprevalenceofmyopiaamonghighschoolandcollegestudentsismorethan70percent,andthesituationiscontinuouslyworsening,headded.SunZhu,directorwiththeChinaYouthDevelopmentServiceCenter,saidlonghoursofstudy,lackofexercise,especiallyoutdooractivities,andattachmenttoelectronicscreenaresomeofthetopreasonsbehindthewidespreadmyopiaproblems,andtheincreaseofyoungerpatients,althoughChinahasmadegreatprogressinimprovingthehardwarefacilitiesandlightingconditionsinschools.Hesaidhehopesthesurveywillalertpeopleonthethreateningsituation,andalsohelpmedicalexpertstobetterguidelineparentsandchildrensbehaviorstocareforeyes,basedonthetimelyandfactualstatisticsthesurveycollected.Aseriesoffollow-upeventswillalsobeheldinsixcities,includingJinaninShandongprovince,andChangshainHunanprovince,toeducatepeopleonhowtocareforeyesafterthesurveystarted. More and more Chinese are jogging to get and stay fit. Most joggers will run alone, but some will jog with friends or even coaches. The Beijing Olympic Forest Park is one of the most popular spots. It is packed with runners in the afternoon, usually after work, and at weekends. Jogging after a whole days work can be a way to unwind. It is true for He Wenjun, whose workplace is right near the forest park. She discovered her athletic aptitude at the age of seven, when she began playing tennis. Then she turned to jogging, and she has persevered with it for 10 years. I come to jog almost every day after work, whenever I have time. I think jogging is good for my health. And for me, a girl, it helps me to achieve to eternal goal, to keep slim, He said. He says she prefers jogging alone, so she can focus better. She loves jogging, even during Beijings smoggy days. If the smog is severe, Ill hold off jogging for a while. But if I really want to get away from work, Ill wear a mask, or just ignore the smog and go on jogging, she said. For those who are not so athletic, jogging also has its attractions. Wu Houbin started roughly two years ago, when he was severely obese. Within two months, his weight dropped from 82 kilograms to 64 kilograms. The success helped him recover health, and he has made friends. Jogging does not require speed, but stamina. If you want to stick to the game, you need to take it as a mission, a habit, and then eventually a hobby, Wu said. But persevering with jogging is not always easy. Jogging is beneficial for heart-lung function, and the skeletal system, but doctors say improper technique damages the joints, especially the knees and ankles. Excessive jogging wears down the body, and harms ones health. Poor technique harms the muscles and skeleton, and can cause secondary injures. And, jogging in a bad environment can cause circulation problems, said Lu Zhiyong, attending doctor, Sports Hospital, General Admin. of Sports.Dr. Lu has some tips for joggers, and those who are thinking of taking it up.He says people should first evaluate their physical condition, to decide the amount and type of exercise, and jogging in the open air is better running on a treadmill. Conrad Macao will run special programs for the entire month of October this year to support the Hong Kong Cancer Funds Pink Revolution, an annual campaign that seeks to raise awareness and funds for breast cancer research, the hotel recently announced. A major sponsor of the campaign for three years in a row, Conrad Macao has pledged to donate a portion of its proceeds to the fund, including those collected from selling its popular bear and rubber duck merchandise, in limited pink editions. Pink is the theme for floral arrangements and decorations in the hotel in October. In addition, on Fridays that month, members of the management and staff will add something pink to their uniforms, regular e-mails and handwritten guest cards. Breast cancer is the most prevalent type of cancer among women in Hong Kong and is also of significant concern in Macao and around the world, regardless of age, general manager Bede Barry said in a statement. The hotel has enlisted the support of a number of international and local celebrities to help drive the cause, including South Korean actor Jung Suk Won. This year, a lucky guest who stays at the hotel during the campaign and also donates money for the cause will win a necklace specially designed by Arte Madrid, a Spanish jewelry brand. Women should strive to balance quality of work and life, and have children at a young age. As a doctor I see much suffering because of this imbalance. Xiao Ying, 36, works for a large law firm. Her job means lots of travel, a heavy caseload and a large number of meetings, so her schedule is hectic. She has been pregnant twice, but had abortions on both occasions because she felt her busy, high-pressure career wasnt conducive to raising a child. She planned to have children once she had achieved career success so she could provide a stable environment for her baby. After 10 years, Xiao felt the time was right, and she tried to conceive for about six months, but was unable to become pregnant. I conducted comprehensive physical checkups. The results suggested that Xiaos ovarian function has declined and a bilateral oviduct blockage has occurred, meaning that in vitro fertilization is her only option. So, after reducing her workload, Xiao underwent IVF treatment. Luckily, she got pregnant after just one treatment cycle, but later experienced vaginal bleeding, and rushed to the hospital. I told her to rest and carry on with the treatment (to prevent a miscarriage). As Zubin Zarthoshtimanesh, a yoga guru, demonstrated postures at a high school stadium in Dujiangyan city, Sichuan province, some 200 students sat on colorful mats watching him closely. It was part of the first India-China international yoga festival in the provincial capital, Chengdu, in Southwest China, where more than 1,000 fans from home and abroad gathered for lessons from 20 leading practitioners of the ancient Indian regimen, from June 17-21. Yoga is to India what perhaps tai chi is to China. Many countries other than India and China, also held the first World Yoga Day on June 21, after the United Nations last year agreed to Indian Prime Minister Narendra Modis suggestion that a special day be marked in the year for the global practice of the stress-busting regimen, which is already a multibillion dollar industry in the United States. Modi said in the speech: Yoga is an invaluable gift of Indias ancient tradition. . It is not about exercise but to discover the sense of oneness with yourself, the world and nature.Zarthoshtimanesh, who runs his own yoga center, Iyengar Yogabhyasa, in the Indian financial capital of Mumbai, says: Chinese students are very respectful. For me, all students are the same no matter what their race is and where they come from. The most important thing is to remove mental walls. Zarthoshtimanesh learned yoga from the late Yogacharya Iyengar and traveled with the master to different countries. Consulate General of India in southern Chinas Guangzhou city organized the Chengdu festival. A terrifying wooden bridge in China has just been replaced by a glass one. This, the worlds longest glass-bottomed walkway, is located in Shiniuzhai Geopark in Hunan, and spans 300m (984f) and is 180m (590f) above ground. The floor is made of double-layered glass that is 24mm (0.94in), and is reportedly 25 times stronger than regular window glass. The bridge had previously been made of wood, linking the two peaks of Stone Buddha Mountain, but 11 engineers working 12 hours a day converted it to glass. Previously, you were considered brave if you steeled yourself to cross the wooden walkway; now, with its glass bottom, the bridge is already being referred to as “hero bridge.” The popular chains longtime executive chef embraces mapo tofu and other exotic elements for the burger-driven menu, Liu Zhihua reports.At the age of 37, Shanghai native Lin Zhengu has become a star as a Western cuisine chef in the Chinese metropolis, a city that has a wealth of Western restaurants. The executive chef of Blue Frog Bar and Grill, a popular burger-based restaurant and bar chain on the Chinese mainland, Lin is the soul of the restaurant franchises team of more than 550 chefs. Many of the franchises most popular dishes ar