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Integrate the rational functionsx/((x+1...

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

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` int (x)/((x+1)(x+2)) dx =I (" say ")`
` " Let " (x)/((x+1)(x+2)) =(A)/((x+1))+(B)/((x+2))`
`rArr (x)/((x+1)(x+2))=(A(x+2)+B(x+1))/((x+1)(x+2))`
` rArr x=A (x+2)+B(x+1)`
`rArr x=-1 " then "-1= A(-1+2)+0`
`A=-1`
`rArr x=-2 " then "-2=0 +B (-1)`
`B=2`
`:. I = int (x)/((x+1)(x+2))dx `
`= int ((-1))/(x+1)dx+ int (2)/(x+2)dx`
`=- log (x+1) +2 log (x+2)+c`
` =- log (x+1)+log (x+2)^(2)+c` ltb rgt `=log |(x+2)^(2)/(x+1)|+c`
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