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

Explanation

`int(2x-3)/((x^(2)-1)(2x+3))dx`
`=int(2x-3)/((x-1)(x+1)(2x+3))dx`
`" Let " (2x-3)/((x-1)(x+1)(2x+3))`
`=(A)/((x-1))+(B)/((x+1))+(C )/((2x+3))`
`rArr (2x-3)/((x-1)(x+1)(2x+3))`
`=(A(2x+3)(x+1)+B(x-1)(2x+3)+C(x-1)(x+1))/((x-1)(x+1)(2x-3))`
`rArr 2x -3 = A(2x+3) (x+1) +B(x-1)(2x+3)`
`+C(x-1)(x+1)`
`X=1 " then " 2-3=A (5)(2) +0+0 rArr A=-(1)/(10)`
`x=-1 " then " -2-3=0+B(-2)(1)+0 rArr B=(5)/(2)`
`rArr x=- (3)/(2) " then " -3-=0 +0+C ((5)/(2)) (-(1)/(2))`
`rArr C=-(24)/(5)`
`:. A =-(1)/(10), B=(5)/(2) " and " C=- (24)/(5)`
`:. int (2x-3)/((x^(2)-1)(2x+3))dx`
`= int ((-1))/(10(x-1))dx+(5)/(2)int(1)/(x+1)dx`
`-(24)/(5)int(1)/(2x+3)dx`
`=-(1)/(10)log|x-1|+(5)/(2)log|x+1|`
`-(24)/(5)(log |2x+3|)/(2)+C_(1)`
`=(5)/(2) log |x+1|-(1)/(10)log|x-1|`
`-(12)/(5)log|2x+3|+C_(1)`