A-level有理函数积分习题介绍

(其中p^2-4q<0,...,r^2-4s<0.).

那么真分式P(x)/Q(x)可以分解成如下部分分式之和：

P(x)/Q(x)=A1/(x-a)^α+A2/(x-a)^(α-1)+...+A[α]/(x-a)+...++B1/(x-b)^β+B2/(x-b)^(β-1)+...+B[β]/(x-b)+(M1x+N1)/(x^2+px+q)^λ+...+(M[λ]x+N[λ])/(x^2+px+q)+......+(R1x+S1)/(x^2+rx+s)^μ+...+(R[μ]x+S[μ])/(x^2+rx+s).

x/[(x+1)(x+2)(x+3)]=A/(x+1)+B/(x+2)+C/(x+3),

x=A(x+2)(x+3)+B(x+1)(x+3)+C(x+1)(x+2).

令x=-1,得A=-1/2,

令x=-2,得B=2,

令x=-3,得C=-3/2,

x/[(x+1)(x+2)(x+3)]=(-1/2)*1/(x+1)+2/(x+2)-(3/2)*1/(x+3),

或由x=(A+B+C)x^2+(5A+4B+3C)x+(6A+3B+2C),

比较系数得

A+B+C=0,5A+4B+3C=1,6A+3B+2C=0,

解出A,B,C.

3/(x^3+1)=1/(x+1)(x^2-x+1)=A/(x+1)+(Mx+N)/(x^2-x+1),

3=A(x^2-x+1)+(Mx+N)(x+1).

令x=-1,得A=1,

(Mx+N)(x+1)=3-A(x^2-x+1)=-x^2+x-2=-(x-2)(x+1),

Mx+N=-x+2,M=-1,N=2.

3/(x^3+1)=1/(x+1)-(x-2)/(x^2-x+1).