General antiderivative of sin(x) + 3cos(x)

Apply linearity to split the integral: $\int (\sin x + 3\cos x)\,dx = \int \sin x\,dx + 3\int \cos x\,dx$. Recall that $\int \sin x\,dx = -\cos x$ and $\int \cos x\,dx = \sin x$, since differentiating these recovers $\sin x$ and $\cos x$ respectively.

Substituting these results gives $-\cos x + 3\sin x$. Since any constant added to an antiderivative still has derivative equal to $f(x)$, we must include an arbitrary constant $C$, giving the general antiderivative $F(x) = -\cos x + 3\sin x + C.$

Because the derivative of any constant is zero, adding a constant to negative cosine plus three sine does not change its derivative. So the derivative still equals sine plus three cosine, meaning every such function is a valid antiderivative. This is why the general antiderivative must include an arbitrary constant, often called C.