1、Integration of Rational Fractions1Rational Functions 2Every rational function may be represented in the form of a rational fraction:10111011()(,)()nnnnmmmma xa xaxaP xm nNQ xb xb xbxbNote We assume these polynomials do not have common roots.Note If n m,the faction is called proper rational fraction
2、真真分式分式,otherwise,i.e.n m,the faction is called improper rational fraction 假分式假分式。Integration of Proper Rational Fraction3(Types of partial fraction)Proper rational fraction of the form:I.(A and a are constants);II.III.(the roots of the denominator are complex);IV.the roots of the denominator are com
3、plex)are called the partial fractions of types I,II,III and IV.Axa2(,);()kAkNkxa2AxBxpxq22(,()kAxBkNkxpxqIntegration of Proper Rational Fraction4Note If the denominator 22()()()()(),Q xxaxbxpxqxrxsthen 1211211122221211222212()()()()()()()()()()AAAP xQ xxaxaxaBBBxbxbxbM xNM xNM xNxpxqxpxqxpxqR xSR xS
4、R xSxrxsxrxsxrxs Integration of Proper Rational Fraction5 Represent the proper rational fraction in the form of a sum of partial rational fraction.2356xxxSolution2335632()()xxxxxx235632,xABxxxxAssumethen233()().A xB xxSo,1233.ABABSolving the system we find65,.AB Therefore23655632.xxxxxFinish.Integra
5、tion of Proper Rational Fraction6 Represent the proper rational fraction in the form of a sum of partial rational fraction.43212221xxxxSolution4322212221111,()ABCxDxxxxxxxAssumethen22211111()()()()().A xB xxCxDxSo,120201,BCABCDBCDABDi.e.,Therefore432221112221212121.()()()xxxxxxxxFinish.1 21 21 20.AB
6、CD Integration of Proper Rational Fraction7 I.(A and a are constants);Axaln.AdxAxaCxa11().()()()kkkAAdxAxadxCxakxa II.2(,);()kAkNkxaIntegration of Proper Rational Fraction8 III.2AxBxpxq222222222222222222422242224242244224()()(/)(/)()()(/)/()()()()()/()()ln()arctau xpAxBAxBdxdxxpxqxpqpA xpBApdxxpqpA
7、xpBApdxdxxpqpxpqpAd uBApduuqpuqpABApxpxqqp 224n.xpCqpIntegration of Proper Rational Fraction9 IV.2()kAxBxpxq2211,()()()kkkAxBdxICxpxqkxpxqwhere12221221232121142;()()()arctan;(;).kkktkIImktmmktppImqtxmmIntegration of Proper Rational Fraction10Compute the following integrals:2243223543131);2);256123);
8、4);22212385);6).3xdxdxxxxxxdxdxxxxxxxxxxdxdxxxx Integration of Proper Rational Fraction11211);2dxxx Solution211131212dxxxdxxx 1ln|1|ln|2|.3xxC11ln.32xCx Integration of Proper Rational Fraction12232)56xdxxx Solution26536235xdxxdxxxx 65326ln|3|5ln|2|.dxdxxxxxCIntegration of Proper Rational Fraction134
9、3213)2221dxxxxx Solution32422112(1)2(1)2(1)12221xdxxdxxxxxxx 222112(1)2(1)2(1)111ln|1|ln|1|.2(1)24xdxdxdxxxxxxCx Integration of Proper Rational Fraction14224)23xdxxx Solution2221(23)3223223xxxdxxxdxxx 222111(23)322323d xxdxxxxx2221ln(23)32(1)(2)dxxxx 2221(1)ln(23)32(1)(2)d xxxx 2131ln(23)arctan.222x
10、xxC Integration of Proper Rational Fraction1535)3xdxx Solution32(39)(3)2733xxxxdxdxxx 23227(39)313927ln|3|.32xxdxdxxxxxxC Integration of Proper Rational Fraction1654386)xxdxxx Solution542323338(1)()8xxxxxxxxdxdxxxxxxx228(1)(1)(1)xxxxdxdxx xx2834(1)11xxdxdxdxdxxxx32118ln|3ln|1|4ln|1|.32xxxxxxCQuadrat
11、ure Problems for elementary fundamental functions17By the previous examples,we have seen that quadratures are much more difficult than differentiations.When integrands are continuous,their integral must exist,but their computation sometimes requires skill,and sometimes may nor even be expressible by
12、 elementary functions.For instance,the integrals:24sin,1xxdxe dxdxxx seems very simple,and the integrands are all continuous.All of these integrals exist,but we can not express them in terms of elementary functions.In general,we have known that for any rational function and any rational trigonometric function,their integrals can be expressed by elementary functions.
