Evaluate: `int(x+1)/((x-1)sqrt(x+2))\ dx`

Answer

`2sqrt(x+2)+(2)/(sqrt(3))log|(sqrt(x+2)-sqrt(3))/(sqrt(x+2)+sqrt(3))|+C`

Explanation

`I =int(x+1)/((x-1)sqrt(x+2))dx`
`" Let " x+2=t^(2) `
`:. dx=2tdt `
`:. I=int(t^(2)-1)/((t^(2)-3)t)2tdt `
`=2int(t^(2)-3+2)/((t^(2)-3))dt `
`=2int(1+(2)/((t^(2)-3)))dt `
`=2t +(2)/(sqrt(3))log|(t-sqrt(3))/(t+sqrt(3))|+C`
`=2sqrt(x+2)+(2)/(sqrt(3))log|(sqrt(x+2)-sqrt(3))/(sqrt(x+2)+sqrt(3))|+C`