1. The problem is to evaluate the integral $$\int \frac{x}{x^2+1} \, dx$$. 2. We use the substitution method. Let $$u = x^2 + 1$$. 3. Then, $$du = 2x \, dx$$, so $$\frac{du}{2} = x \, dx$$. 4. Substitute into the integral: $$\int \frac{x}{x^2+1} \, dx = \int \frac{1}{u} \cdot \frac{du}{2} = \frac{1}{2} \int \frac{1}{u} \, du$$. 5. The integral of $$\frac{1}{u}$$ with respect to $$u$$ is $$\ln|u| + C$$. 6. Therefore, $$\frac{1}{2} \int \frac{1}{u} \, du = \frac{1}{2} \ln|u| + C$$. 7. Substitute back $$u = x^2 + 1$$ to get the final answer: $$\frac{1}{2} \ln|x^2 + 1| + C$$. Final answer: $$\int \frac{x}{x^2+1} \, dx = \frac{1}{2} \ln|x^2 + 1| + C$$.