1. The problem is to find the integral of the function $d(e^{g(x)})dx$. 2. This expression suggests the differential of $e^{g(x)}$ with respect to $x$, which is $d(e^{g(x)}) = e^{g(x)} g'(x) dx$ by the chain rule. 3. Therefore, the integral of $d(e^{g(x)})$ with respect to $x$ is simply $e^{g(x)} + C$, where $C$ is the constant of integration. 4. In summary, $$\int d(e^{g(x)}) = e^{g(x)} + C$$ 5. This is because integrating a differential $dF(x)$ returns the original function $F(x)$ plus a constant.