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.

  • $\begingroup$ How can the value of $\log(\phi(x))$ depend only on $\frac{\partial \log(\phi(x))}{\partial x_1}$ ? What makes $x_1$ special ? What if $\phi$ has no dependence on $x_1$ so the integral is always zero ? $\endgroup$ – gandalf61 Apr 12 at 13:33
  • $\begingroup$ Maybe this is what the constant $\alpha$ is there for? I don't know, I'm just trying to guess. $\endgroup$ – STF Apr 12 at 13:36

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