f(x) = (x^3 - x - 2) / [(x-1)(x^2 + 1)]
(i) Express f(x) in form of A + B/(x-1) + (Cx + D)/(x^2 + 1) where A,B,C and D are constants
(ii) Find integrate f(x) from 2 to 3
integration Q (2)
2012-09-01 9:43 pm
回答 (3)
2012-09-04 1:11 am
✔ 最佳答案
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參考: miraco
2012-10-23 12:10 am
To 001:
In the 2nd line, it should be (Cx + D)(x - 1) not Cx + D.
In the 2nd line, it should be (Cx + D)(x - 1) not Cx + D.
2012-09-01 10:13 pm
A + B/(x-1) + (Cx+D)/ (x^2+1)
= 1/(x-1)(x^2+1) [A(x-1)(x^2+1) + B(x^2+1) + Cx(x-1)+ D(x-1)]
= 1/(x-1)(x^2+1) [A(x^3 - x^2 +x -1) + Bx^2 + B + Cx^2 -Cx+ Dx-D]
= 1/(x-1)(x^2+1) [Ax^3 - Ax^2 +Ax -A + Bx^2 + B + Cx^2 -Cx+ Dx-D]
= 1/(x-1)(x^2+1) [Ax^3 +(C-A)x^2 +(A-C+D)x +(-A +B -D)]
By comparing coefficient with f(x), we obtain:
A=1
C-A=0
A-C+D=-1
-A+B-D=-2
so A=1, C-1=0=>C=1
1-1+D=-1 => D=-1
-1+B-(-1)=-2 =>B=-2
= 1/(x-1)(x^2+1) [A(x-1)(x^2+1) + B(x^2+1) + Cx(x-1)+ D(x-1)]
= 1/(x-1)(x^2+1) [A(x^3 - x^2 +x -1) + Bx^2 + B + Cx^2 -Cx+ Dx-D]
= 1/(x-1)(x^2+1) [Ax^3 - Ax^2 +Ax -A + Bx^2 + B + Cx^2 -Cx+ Dx-D]
= 1/(x-1)(x^2+1) [Ax^3 +(C-A)x^2 +(A-C+D)x +(-A +B -D)]
By comparing coefficient with f(x), we obtain:
A=1
C-A=0
A-C+D=-1
-A+B-D=-2
so A=1, C-1=0=>C=1
1-1+D=-1 => D=-1
-1+B-(-1)=-2 =>B=-2
參考: ME
收錄日期: 2021-04-13 18:57:15
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