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

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

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

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