`(2x-3)/((x^(2)-1)(2x+3))`

Explanation

`int(2x-3)/((x^(2)-1)(2x+3))dx`
`" "=int(2x-3)/((x-1)(x+1)(2x+3))dx`
माना `" "(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,"तो ",2-3=A(5)(2)+0+0,rArr.,A=-(1)/(10)),(x=-1,"तो ",-2-3=0+B(-2)(1)+0,rArr.,B=(5)/(2)):}`
`rArr x=-(3)/(2)" तो "-3-3=0+0+C(-(5)/(2))(-(1)/(2))`
`rArr C=-(24)/(5)`
`therefore" "A=-(1)/(10), B=(5)/(2)" तथा "C=-(24)/(5)`
`therefore" "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)`