Find `int` `(x^2-5x)/((x-1)(x+1)^2) dx`

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The integral `int (x^2 - 5x)/((x - 1)(x + 1)^2) dx` has to be determined.

`int (x^2 - 5x)/((x - 1)(x + 1)^2) dx`

Let `A/(x - 1) + B/(x + 1) + C/(x + 1)^2 = (x^2 - 5x)/((x - 1)(x + 1)^2)`

=> `A(x+1)^2 + B(x - 1)(x + 1) + C(x - 1) = x^2 - 5x`

=> `Ax^2 + 2Ax + A + Bx^2 - B + Cx - C = x^2 - 5x`

=> A + B = 1, 2A + C = -5 and A - B - C = 0

Use the system of equations to solve for A, B, C.

A + B = 1 => B = 1 - A and 2A + C = -5 => C = -5 - 2A

A - B - C = A - 1 + A + 5 + 2A = 0

=> 4A = -4

=> A = -1

B = 2

C = -3

`int (x^2 - 5x)/((x - 1)(x + 1)^2) dx`

= `int -1/(x - 1) + 2/(x + 1) - 3/(x + 1)^2 dx`

= `-1*ln(x - 1) + 2*ln(x + 1) + 3/(x + 1)`

**The integral `int (x^2 - 5x)/((x - 1)(x + 1)^2) dx = -1*ln(x - 1) + 2*ln(x + 1) + 3/(x + 1) + C` **

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