How do I solve this integral by completing the square and making a substitution?

Question

The problem is: the integral of dx / x^2 + x + 1

I have to complete the square on the quadratic and make a substitution and then evaluate. How do I go about doing this without changing the value of the integrand??

Much thanks :)

Answer 1

Change the integral to that of dx/(Y^2 + A), where Y = X + 1/2, and you can calculate the constant A.

But that's the same as the integral of dY/(Y^2 + A).

Hopefully you can solve that directly.

If not, let B be a constant such that Z = - B^2, and look at the relationship between 1/(Y^2 - B^2) and [1/(Y-B) - 1/(Y+B)]. (One's a constant multiple of the other.)

You should be able to fill in the blanks.