I am trying to decompose the following rational function: <mstyle displaystyle="true" scriptleve

Lydia Carey

Lydia Carey

Answered question

2022-06-21

I am trying to decompose the following rational function: 1 ( x 2 1 ) 2 in partial fractions (in order to untegrate it later).
I have notices that ( x 2 1 ) 2 = ( x + 1 ) 2 ( x 1 ) 2
Therefore A , B , C , D s.t:
1 ( x 2 1 ) 2 = 1 ( x 1 ) 2 ( x + 1 ) 2 = A x + B ( x + 1 ) 2 + C x + D ( x 1 ) 2
We then have
A x + B = 1 ( x 1 ) 2 | x = 1 B A = 1 / 4
Same thing for C x + D:
C x + D = 1 ( x + 1 ) 2 | x = 1 C + D = 1 / 4
How do I find A, B, C, D from here?

Answer & Explanation

kpgt1z

kpgt1z

Beginner2022-06-22Added 23 answers

We have
1 = ( x 1 ) 2 ( A x + B ) + ( x + 1 ) 2 ( C x + D )
You already have
B A = C + D = 1 4
Substituting x = 0 gives
1 = B + D
Substituting x = 2 gives
1 = 2 A + B + 9 ( 2 C + D )
Solving these gives
A = 1 4 , B = 1 2 , C = 1 4 , D = 1 2
Leah Pope

Leah Pope

Beginner2022-06-23Added 7 answers

Consider:
1 ( x 2 1 ) 2 = 1 ( x + 1 ) 2 ( x 1 ) 2 = A x + 1 + B ( x + 1 ) 2 + C x 1 + D ( x 1 ) 2
Try to decompose the above

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