Evaluate : `int(3x+1)/((x-1)^(3)(x+1))dx`

Explanation

Let `(3x+1)/((x-1)^(3)(x+1))= A(x+1)+B/(x-1)+C/(x-1)^(2)+D/(x-1)^(3)`…………(i)
Multiplying both sides by (x+1) and then putting `x=-1`, we get
`A=(-2)/(-2)^(3)=1/4`
Multiplying both sides by `(x-1)^(3)` and then pulling x=1, wee get `D=4/2-2`
From (i), we get
`3x+1=A(x-1)^(3)+B(x-1)^(2)(x+1)+C(x-1)(x+1)+D(x+1)`
`1=-A+B-C+D`
`rArr 1=-1/4+B-C+2 rArr B-C=-3/4`
Putting x=2, we get
`7=A+3B+3C+3D`
`rArr 7=1/4+3B+3C+6 rArr 3B+3C=3/4 rArr B+C=1/4`
Solving B+C`=1/4` and `B-C=-3/4`, we get `B=-1/4, C=1/2`
Substituting the values of A,B,C and D in (i), we get
`rArr (3x+1)/((x-1)^(3)(x+1))=1/4.(1/(x+1))-1/(4(x-1))+1/(2(x-1)^(2))+2/(x-1)^(3)`
`rArr int(3x+1)/((x-1)^(3)(x+1))dx=1/4int1/(x+1)dx-1/4int1/(x-1)dx+1/2int1/(x-1)^(2)dx+2int1/(x-1)^(3)`dx
`=1/4ln|x+1|-1/4ln|x-1|-1/(2(x-1))-1/(2(x-1))-1/(x-1)^(2)+C`