The problem presents a differential equation $\frac{dV}{dx} = -w(x)$. We are asked to solve this differential equation for $V$.

1. Problem Description

The problem presents a differential equation $\frac{dV}{dx} = -w(x)$. We are asked to solve this differential equation for $V$.

2. Solution Steps

The given differential equation is

$\frac{dV}{dx} = -w(x)$.

To solve for $V$, we need to integrate both sides of the equation with respect to $x$:

$\int \frac{dV}{dx} dx = \int -w(x) dx$

$V = -\int w(x) dx$

Let $W(x)$ be the antiderivative of $w(x)$. Then $\int w(x) dx = W(x) + C$, where $C$ is the constant of integration. Therefore,

$V(x) = -W(x) + C$,

$V(x) = -\int w(x) dx$.

3. Final Answer

$V(x) = -\int w(x) dx$