To find the integral of $-\frac{2x}{x^2 - x + 1}$, we can use the technique of substitution. Let's proceed step by step.

1. Substitution: Let $u = x^2 - x + 1$. Then, the derivative $du = (2x - 1)dx$.
2. Rearranging for $dx$: From the above, we have $dx = \frac{du}{2x - 1}$.
3. Express $x$: We can also express $x$ in terms of $u$ from the substitution, but for our integral, we will express $dx$ in terms of $du$ after manipulating the integrand.

Next, we express $-\frac{2x}{u}$ in terms of $u$:

4. Writing the integral: $\int -\frac{2x}{u} \cdot \frac{du}{2x - 1}$

However, it's easier to use the partial fraction method or a different substitution method directly for our integral.

Direct Integration:

1. Rewrite the integral: $\int -\frac{2x}{x^2 - x + 1} \, dx$

2. Let’s consider a simple manipulation: We can split the integral as follows: $= \int -\frac{2x - 1 + 1}{x^2 - x + 1} \, dx = \int -\frac{2x - 1}{x^2 - x + 1} \, dx - \int \frac{1}{x^2 - x + 1} \, dx$

3. The first integral: The first part $-\int \frac{2x - 1}{x^2 - x + 1} \, dx$ can be integrated directly using substitution $u = x^2 - x + 1$: $-\ln|u| + C_1 = -\ln|x^2 - x + 1| + C_1$

4. The second integral: For $\int \frac{1}{x^2 - x + 1} \, dx$, we complete the square in the denominator: $x^2 - x + 1 = \left(x - \frac{1}{2}\right)^2 + \frac{3}{4}$ This leads to: $\int \frac{1}{\left(x - \frac{1}{2}\right)^2 + \left(\frac{\sqrt{3}}{2}\right)^2} \, dx = \frac{2}{\sqrt{3}} \tan^{-1}\left(\frac{2x - 1}{\sqrt{3}}\right) + C_2$

Final Result:

Combining both integrals, we have: $\int -\frac{2x}{x^2 - x + 1} \, dx = -\ln|x^2 - x + 1| - \frac{2}{\sqrt{3}} \tan^{-1}\left(\frac{2x - 1}{\sqrt{3}}\right) + C$

Where $C$ is the constant of integration.

Would you like to see more details about any specific step? Here are some related questions you might find useful:

1. How do you perform substitution in integrals?
2. What is the process for completing the square?
3. Can you explain the derivation of the arctangent integral?
4. What are some common techniques for integration?
5. How does the natural logarithm relate to integrals of rational functions?

Tip: Always check if the integrand can be simplified or rewritten before integrating, as this often makes the process easier!