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

Explanation

माना `(x)/((x-1)^(2)(x+2))`
`=(A)/(x-1)+(B)/((x-1)^(2))+(C)/(x+2)" …(1)"`
`rArr x=A(x-1)(x+2)+B(x+2)+C(x-1)^(2)" …(2)"`
`{:(x=1,"तो ",1=0+B(3)+0,rArr.,B=(1)/(3)),(x=-2,"तो ",-2=0+0+C(9),rArr.,C=-(2)/(9)):}`
`x^(2)` के गुणांक को बराबर रखने पर
`0=A+C" "rArr" "A=-C=(2)/(9)`
`therefore int(x)/((x-1)%(2)(x+2))dx`
`int{((2)/(9))/(x-1)+((1)/(3))/((x-1)^(2))+((-(2)/(9))/(x+2))}dx`
[समीकरण (1 ) से]
`=(2)/(9)int(1)/(x-1)dx+(1)/(3)int(1)/((x-1)^(2))dx-(2)/(9)int(1)/(x+2)dx`
`=(2)/(9)log|x-1|+(1)/(3)((x-1)^(-2+1))/((-2+1))-(2)/(9)log|x+2|+C`
`=(2)/(9)log|(x-1)/(x+2)|-(1)/(3)((1)/(x-1))+C`