Integrate the functions`(x^2+x+1)/((x+1)^2(x+2)`

Explanation

`int (x^(2) +x+1)/((x+1)^(2)(x+2))dx`
`" Let " (x^(2) +x+1)/((x+1)^(2)(x+2))=(A)/(x+1)+(B)/((x+1)^(2))+(C)/((x+2))`
`rArr x^(2) +x+1 =A (x+1) (x+2) " "......(1)`
`+B(x+2)+C(x+1)^(2)`
x=- 1 then 1-1 +1 =0 +B (-1+2) +0` rArr` B=1
x=-2 then 4-2 +1 -0 +0 +C(1) `rArr ` c=3
Equation the coefficients of `x^(2)`
` 1=A +C rArr A=-2`
`:. (x^(2)+x+1)/((x+1)^(2)(x+2))=(-2)/(x+1)+(1)/((x+1)^(2))+(3)/((x+3))`
`:. int (x^(2)+x+1)/((x+1)^(2)(x+2))dx`
`=-2 int (dt)/(x+1) +int (dx)/((x+1)^(2))+3int (dx)/((x+2))`
`=-2 log |x+1| -(1)/((x+1))+3log |x+2|+C`