How do you integrate x^2 / (x-1)^3 using partial fractions?

1 Answer
Roy E.
Dec 7, 2016

ln(x-1)-2/(x-1)-1/(2(x-1)^2)+C

Explanation:

Assume that there exist A, B and C such that:
x^2/(x-1)^3-=A/((x-1))+B/(x-1)^2+C/(x-1)^3.
To find A, B and C substitute any three different numbers other than -1 and solve the resulting set of three linear equations. For example, set x=0, 2 and -1:
0=A/-1+B+C/-1
4=A+B+C
-1/8=A/-2+B/4-C/8
giving A=1, B=2 and C=1.
So the required integral is:
int1/(x-1)+2/(x-1)^2+1/(x-1)^3dx
which gives the answer above.

Alternatively, instead of using partial fractions, substitute u=x-1 to get int1/u+2/u^2+1/u^3du.

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