1. The problem is to find the integral of the function $f(x) = \sin x + \cos x$. 2. The integral of a sum of functions is the sum of their integrals, so we use the formula: $$\int (\sin x + \cos x) \, dx = \int \sin x \, dx + \int \cos x \, dx$$ 3. Recall the basic integrals: - $\int \sin x \, dx = -\cos x + C$ - $\int \cos x \, dx = \sin x + C$ 4. Applying these, we get: $$\int (\sin x + \cos x) \, dx = -\cos x + \sin x + C$$ 5. Therefore, the integral of $f(x)$ is: $$\boxed{-\cos x + \sin x + C}$$ This means the antiderivative of $\sin x + \cos x$ is $-\cos x + \sin x$ plus a constant of integration $C$.