`(3x+5)/(x^(3)-x^(2)-x+1)`

Explanation

`int(3x+5)/(x^(3)-x^(2)-x+1)dx`
`" "=int(3x+5)/((x^(2)1)(x-1))dx`
`" "=int(3x+5)/((x-1)(x+1)(x-1))dx`
`" "=int(3x+5)/((x-1)^(2)(x+1))dx`
माना `" "(3x+5)/((x-1)^(2)(x+1))=(A)/(x-1)+(B)/((x-1)^(2))+(C)/(x+1)`
`rArr" "(3x+5)/((x-1)^(2)(x+1))`
`" "=(A(x-1)(x+1)+B(x+1)+C(x-1)^(2))/((x-1)^(2)(x+1))`
`rArr 3x+5=A(x-1)(x+1)+B(x+1)+C(x-1)^(2)`
`{:(x=1,"तो ",3+5=0+B(2)+0,rArr.,B=4),(x=-1,"तो ",-3+5=0+0+C(-2)^(2),rArr.,C=(1)/(2)):}`
`rArr" "x^(2)` के गुणकों को समान रखने पर,
`" "0=A+C`
`rArr A=-C=-(1)/(2)`
`therefore int(3x+5)/((x-1)^(2)(x+1))dx`
`" "=int(_(1)/(2))/(x-1)+(4)/((x-1)^(2))+((1)/(2))/(x+1)dx`
`" "=-(1)/(2)int(1)/(x-1)dx+4int(1)/((x-1)^(2))dx+(1)/(2)int(1)/(x+1)dx`
`=-(1)/(2)log|x-1|+4((x-1)^(-2+1))/((-2+1))+(1)/(2)log|x+1|+C`
`=(1)/(2)log|(x+1)/(x-1)|-(4)/(x-1)+C`