1. **State the problem:** We need to find the integral $$\int \frac{2x}{x^2+1} \, dx$$. 2. **Recall the formula and rules:** The integral of a function of the form $$\frac{f'(x)}{f(x)}$$ is $$\ln|f(x)| + C$$, where $$C$$ is the constant of integration. 3. **Identify the inner function and its derivative:** Here, the denominator is $$f(x) = x^2 + 1$$. 4. The derivative of $$f(x)$$ is $$f'(x) = 2x$$, which matches the numerator. 5. **Apply the formula:** Since the numerator is the derivative of the denominator, the integral is $$\int \frac{2x}{x^2+1} \, dx = \ln|x^2 + 1| + C$$. 6. **Simplify:** Because $$x^2 + 1 > 0$$ for all real $$x$$, the absolute value can be omitted: $$\ln(x^2 + 1) + C$$. **Final answer:** $$\int \frac{2x}{x^2+1} \, dx = \ln(x^2 + 1) + C$$.