不定积分中有理函数的分解

32()()Vo.l3No.2JOURNALOFGUIYANGCOLLEGE20086NaturalSciences(Quarterly)Jun.2008不定积分中有理函数的分解*赵晓艾(,277160):有理函数的积分是不定积分中的一种重要类型,对这类积分常用的方法是先把有理函数分解为部分分式,然后利用待定系数法和赋值法求解,有理函数积分的重点和难点就是对有理函数进行有效的分解通过实例介绍有理函数的分解技巧,从而可方便地解决这类有理函数的积分问题:有理函数;部分分式;不定积分;分解:O174:A:1673-6125(2008)02-0006-03DecompositionofRationalFunctioninIndefiniteIntegralZhaoXiaoai(MathsandInformationScienceDepartment,ZaozhuangCollege,ZaozhuangShandong277160,China)Abstract:Rationalfunctionintegralisanimportanttypeofindefiniteintegralanditsusualmethodtothisintegralistodividetherationalfunctionintopartfractions,anditcanbesolvedthroughthemethodsofundeterminedcoefficientsandassignmentvalue.Theimportantanddifficultpointistheefficientdecompositionofrationalfunction.Throughexemplarintroductiontothedecompositionskillsofrationalfunction,thequestionofrationalfunctioncanbeconvenientlyanswered.Keywords:rationalfunction;partfractions;indefiniteintegra;ldecomposition0,,,,1,,R(x)=P(x)Q(x)=anxn+an-1xn-1++a0bmxm+bm-1xm-1++b0,(an0,bm0),nm,R(x);n!m,R(x),,,,∀6∀*:2008-03-01:(1979-),,,,2R(x)=P(x)Q(x),R(x),R(x)Q(x),R(x),R(x)=P(x)Q(x),#P(x)Q(x)dx,:(1)Q(x)(x-a)k,R(x)k:A1x-a+A2(x-a)2++Ak(x-a)k,A1,A2,Ak(2)Q(x)(x2+px+q)1,(p2-4q0),R(x)l:B1x+D1x2+px+q+B2x+D2(x2+px+q)2++B1x+D1(x2+px+q)1,B1,D1,B2,D2,Bl,Dl331x2-1x(x+1)3=Ax+Bx+1+Cx+D(x+1)2+Ex+F(x+1)3:(1),,x2-1x(x+1)3=Ax+Bx+1+C(x+1)2+D(x+1)3,(3);,x2-1x(x+1)3x+1,x-1x(x+1)2,x-1x(x+1)2=Ax+Bx+1+C(x+1)2,(4)(3),(4),,,,,32(x-1)(x3+2)x2(x2-x+1)=Ax+Bx2+Cx+Dx2-x+1:,(1)(2),,,,4,3,,?,(1)(2),,R(x),?:(x-1)(x3+2)x2(x2-x+1)=x4-x3+2x-2x4-x3+x2=1-x2-2x+2x2(x2-x+1)=1-1x2+x-1x2(x2-x+1),x-1x2(x2-x+1):x-1x2(x2-x+1)=Ax+Bx2+Cx+Dx2-x+1,∀7∀,33x+14x(x2-1)2=Ax+Bx+Cx2-1+Dx+E(x2-1)2:(2)Q(x)(x2+px+q)1,(p2-4q0),p2-4q0,x2+px+qx2-1:x2-1=(x+1)(x-1),:x+14x(x2-1)2=14x(x+1)(x-1)2=14Ax+Bx+1+Cx-1+D(x-1)2,A,B,C,D,,,,,,,;,,,,:[1].[M].:,2004.(5)Sn=a11-q[(1-q)+(1-q2)+(1-q3)++(1-qn)]=a11-qn-q(1-qn)1-q=a11-qn-q-qn+11-q(q1):1=1,q=2,1+(1+2)+(1+2+4)++(1+2+4++2n-1)=11-2(n-2-2n+11-2)=2n+1-n-21=1,q=3,1+(1+3)+(1+3+9)++(1+3+9++3n-1)=11-3(n-3-3n+11-3)=14(3n+1-2n-3),,,,:[1],.[M].:,2007.∀8∀