Finding factors in order to use U sub

[TheKShaugh]

Homework Statement

$$\displaystyle{\int}\dfrac{x}{\sqrt{x^2+x+1}}~dx =<br /> <br /> \displaystyle{\int}\dfrac{(x+\frac 1 2)-\frac 1 2}{\sqrt{(x+\frac 1 2)^2+(\frac {\sqrt 3} 2)^2}}~dx$$

Homework Equations

Given.

The Attempt at a Solution

The solution is given, but I'm not sure how it was found. Is there a method for finding those factors or is it trial and error/intuition?

 

[Mark44]

Homework Statement

$$\displaystyle{\int}\dfrac{x}{\sqrt{x^2+x+1}}~dx =<br /> <br /> \displaystyle{\int}\dfrac{(x+\frac 1 2)-\frac 1 2}{\sqrt{(x+\frac 1 2)^2+(\frac {\sqrt 3} 2)^2}}~dx$$

Homework Equations

Given.

The Attempt at a Solution

The solution is given, but I'm not sure how it was found. Is there a method for finding those factors or is it trial and error/intuition?

There is a method. They are completing the square in the quadratic in the denominator.
x² + x + 1 = x² + x + (1/4) + (1 - 1/4)
= (x + 1/2)² + 3/4
= (x + 1/2)² + (√(3)/2)²

Then they are working with the numerator to get it as you see it in the expression on the right of what you posted.