1. **State the problem:** Evaluate the integral $$\int \frac{\sin x + \cos x}{\sin x + \cos x} \, dx$$. 2. **Simplify the integrand:** Since the numerator and denominator are the same (and non-zero), the fraction simplifies to 1: $$\frac{\sin x + \cos x}{\sin x + \cos x} = 1$$. 3. **Rewrite the integral:** The integral becomes $$\int 1 \, dx$$. 4. **Integrate:** The integral of 1 with respect to $x$ is $$x + C$$, where $C$ is the constant of integration. 5. **Final answer:** $$\int \frac{\sin x + \cos x}{\sin x + \cos x} \, dx = x + C$$.