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

Explanation

`(2x)/(x^(2)+3x+2)=( 2x)/(x^(2)+2x+x+2)`
`" "=(2x)/(x(x+2)+1(x+2))`
`" "=(2x)/((x+2)(x+1))`
माना `(2x)/((x+2)(x+1))=(A)/((x+2))+(B)/((x+1))`
`rArr" "(2x)/((x+2)(x+1))=(A(x+1)+B(x+2))/((x+2)(x+1))`
`rArr" "2x=A(x+1)+B(x+2)`
`{:(x=-1,"तो ",-2=0+B(1),rArr.,B=-2),(x=-2,"तो ",-4=A(-1)+0,rArr.,A=4):}`
`therefore" "int(2x)/((x+2)(x+1))dx`
`=int(A)/((x+2))dx+int(B)/((x+1))dx`
`=int(4)/((x+2))dx+int((-2))/((x+1))dx`
`=4log|x+2|-2log|x+1|+C`