It seems like you're asking to solve the integral of the function $\frac{2x}{2x^2 + 2}$. I'll go step by step to solve this:

Step 1: Simplify the integrand

The given integral is:

$\int \frac{2x}{2x^2 + 2} \, dx$

First, factor out the common factor of 2 from the denominator:

$\int \frac{2x}{2(x^2 + 1)} \, dx$

Now the integral becomes:

$\frac{1}{2} \int \frac{2x}{x^2 + 1} \, dx$

Step 2: Use substitution

This is a straightforward integral that can be solved using substitution. Let:

$u = x^2 + 1$

Then, the derivative of $u$ with respect to $x$ is:

$du = 2x \, dx$

Thus, the integral becomes:

$\frac{1}{2} \int \frac{du}{u}$

Step 3: Integrate

The integral of $\frac{1}{u}$ is $\ln|u|$. So, we have:

$\frac{1}{2} \ln|u| + C$

Step 4: Substitute back $u$

Since $u = x^2 + 1$, substitute back:

$\frac{1}{2} \ln|x^2 + 1| + C$

Thus, the final answer is:

$\frac{1}{2} \ln(x^2 + 1) + C$

Final Answer:

$\int \frac{2x}{2x^2 + 2} \, dx = \frac{1}{2} \ln(x^2 + 1) + C$

Would you like further details or clarification on any of the steps? Here are some additional questions that could help expand your understanding:

1. What is the process for solving integrals with trigonometric functions, like $\frac{1}{x^2 + 1}$?
2. Can you explain why substitution works in this case and how to recognize when it's necessary?
3. How would you solve an integral with a higher-degree polynomial in the denominator?
4. What are some other methods to solve integrals that cannot be simplified directly?
5. How would you approach an integral involving a rational function with an exponential term?

Tip: When integrating functions of the form $\frac{1}{x^2 + a^2}$, you can use the formula $\int \frac{1}{x^2 + a^2} \, dx = \frac{1}{a} \tan^{-1}\left(\frac{x}{a}\right) + C$, which simplifies the process!