Consider a function $\phi: \mathbb{R}^K \rightarrow (0,1]$.
Suppose that the partial derivative $$ \frac{\partial \log(\phi(x))}{\partial x_1} $$ exists for every vector $x\in \mathbb{R}^K$, where $x_1$ is the first element of the vector $x$.
I want to show that $$ \log(\phi(\bar{x}))=\int_?^? \frac{\partial \log(\phi(x))}{\partial x_1}dx_1+\alpha $$ where $\alpha$ is the constant of integration and $\bar{x}$ is a specific value of $x$.
I think this is just an application of the fundamental theorem of calculus, but I'm lost in understanding what we are doing formally. For example, in which region are we integrating over? Where does the constant of integration come from? For example, if I read here, then there is no constant of integration to consider.