Closed Line Integral 82Bbf2

1. **Problem Statement:** Evaluate the line integral $$\oint x \, dx$$ over a closed curve. 2. **Understanding the Integral:** The symbol $$\oint$$ denotes a line integral over a closed path. Here, the integrand is $$x$$ with respect to $$dx$$. 3. **Key Rule:** For any closed curve, the integral of an exact differential over that curve is zero. Since $$x \, dx$$ is an exact differential of $$\frac{x^2}{2}$$, the integral over a closed path is zero. 4. **Mathematical Explanation:** $$\oint x \, dx = \oint d\left(\frac{x^2}{2}\right) = 0$$ because the integral of an exact differential over a closed loop is zero. 5. **Final Answer:** $$\boxed{0}$$