Question

Evaluate: `int(x-1)/((x+1)sqrt(x^3+x^2+x)) dx`

Answer 1

Correct Answer - `2"tan"^(-1)sqrt(x+(1)/(x)+1)+C`
`I=int((x-1)dx)/((x+1)sqrt(x^(3)+x^(2)+x))`
`=int((x^(2)-1))/((x^(2)+2x+1)sqrt(x^(3)+x^(2)+x))`
`=int((1-(1)/(x^(2)))dx)/((x+(1)/(x)+2)sqrt(x+(1)/(x)+1))`
`"Putting " x+(1)/(x)+1=u^(2),`
`I=int(2udu)/((u^(2)+1)*u)=2 "tan"^(-1)u+C`
`=2"tan"^(-1)sqrt((x+(1)/(x)+1))+C`