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

Answer

`-log|(1)/(x+1)-(1)/(2)+(sqrt(x^(2)+x+1))/(x+1)|+C`

Explanation

`I=int(1)/((x+1)sqrt(x^(2)+x+1))dx`
` "Let " x+1=(1)/(t) " or " dx=-(1)/(t^(2))dt `
`:. I=int(1)/((1)/(t)sqrt(((1)/(t)-1)^(2)+((1)/(t)-1)+1))(-(1)/(t^(2)))dt`
`=-int(dt)/(sqrt((1-t)^(2)+(t-t^(2))+t^(2)))`
`=-int(dt)/(sqrt(t^(2)-t+1))`
` =-int(dt)/(sqrt((t-(1)/(2))^(2)+(3)/(4)))`
`=-log|(t-(1)/(2))+sqrt(t^(2)-t+1)|+C`
`=-log|(1)/(x+1)-(1)/(2)+(sqrt(x^(2)+x+1))/(x+1)|+C`