Integrate the rational functions`(5x)/((x+1)(x^2-4)`

Explanation

`int(5x)/((x+1)(x^(2)-4))dx`
`=int (5x)/((x+1)(x+2)(x-2))dx`
`" Let " (5x)/((x+1)(x-2)(x+2))`
`=(A)/((x+1))+(B)/((x+2))+(C )/((x-2)) " "……..(1)`
`A(x+2)(x-2)+B(x+1)`
`=((x-2)+C(x+1)(x+2))/((x+1)(x+2)(x-2))`
`rArr 5x=A(x+2)(x-2)+B(x+1)(x-2)`
`+C(x+1)(x+2)`
`x=-1 " then " -5=A(1)(-3)+0+0 rArr =(5)/(3)`
`x=-2 " then " -10 =0 +B (-1) (-4) +0 rArr B=-(5)/(2)`
`x=2 " then " 10=0 +0 +C(3) (4) rArr c=(5)/(6)`
`:. int(5x)/((x+1)(x-2)(x+2))dx`
`=int(5)/(3(x-1))dx - int(5)/(2(x-2))dx`
+int(5)/(6(x-2))dx`
=(5)/(3) int(1)/((x+1))dx-(5)/(2)int(1)/((x+2))dx`
`+(5)/(6) int (1)/((x-2))dx`
`=(5)/(2) log |x+1|-(5)/(2)log |x+2|`
`+(5)/(2)log |x-2|+C_(1)`