1. **State the problem:** Evaluate the definite integral $$\int_1^2 \frac{1}{x^2} \, dx$$. 2. **Recall the formula:** The integral of $$x^n$$ with respect to $$x$$ is $$\frac{x^{n+1}}{n+1} + C$$ for $$n \neq -1$$. 3. **Rewrite the integrand:** $$\frac{1}{x^2} = x^{-2}$$. 4. **Integrate:** $$\int x^{-2} \, dx = \frac{x^{-2+1}}{-2+1} + C = \frac{x^{-1}}{-1} + C = -x^{-1} + C = -\frac{1}{x} + C$$. 5. **Evaluate the definite integral:** $$\int_1^2 \frac{1}{x^2} \, dx = \left[-\frac{1}{x}\right]_1^2 = \left(-\frac{1}{2}\right) - \left(-\frac{1}{1}\right) = -\frac{1}{2} + 1 = \frac{1}{2}$$. 6. **Final answer:** $$\boxed{\frac{1}{2}}$$