Evaluate: `int1/((x^2-4)sqrt(x+1))dx`

Answer

`(1)/(4sqrt(3))log|(sqrt(x+1)-sqrt(3))/(sqrt(x+1)+sqrt(3))|-(1)/(2)"tan"^(-1)sqrt(x+1)+C`

Explanation

`"Let " I=int(1)/((x^(2)-4)sqrt(x+1))dx`
`"Putting " x+1=t^(2) " and " dx=2tdt, " we get " `
`I=int(2tdt)/([(t^(2)-1)^(2)-4]sqrt(t^(2)))`
`=2int(dt)/((t^(2)-1-2)(t^(2)-1+2))`
`=2int(dt)/((t^(2)-3)(t^(2)+1))`
`=(2)/(4)int((1)/(t^(2)-3)-(1)/(t^(2)+1))dt`
`=(1)/(4sqrt(3))log|(t-sqrt(3))/(t+sqrt(3))|-(1)/(2)"tan"^(-1)t+C, " where " t=sqrt(x+1)`