How to calculate this kind of integrataion

Question

Context of my problem:
In a continuous-time scenario, consider a factory with a production function $f(\mathbf{x}(t),\mathbf{y})$, where $\mathbf{x}(t)$ (a vector) represents the skills of its workers, and $\mathbf{y}$ represents the technology of this factory. The skills of workers of this firm is time-variant, and it follows $\dot{\mathbf{x}}(t) = \mathbf{g}(\mathbf{x}(t),\mathbf{y})$, while the technology of the factory $\mathbf{y}$ is constant over time. The factory shuts down following a Poisson process with rate $\lambda$, and the discount rate is $r$.
The question is:
How to calculate the factory's expected discounted present value of its production?
My idea is to calculate this integration:
$\int_0^{\infty}\lambda e^{-\lambda T} \int_0^T e^{-rt}f(\mathbf{x}(t),\mathbf{y})dt dT$, where $\dot{\mathbf{x}}(t) = \mathbf{g}(\mathbf{x}(t),\mathbf{y})$.
But I have no idea about how to calculate $\int_0^T e^{-rt}f(\mathbf{x}(t),\mathbf{y})dt$, where $\dot{\mathbf{x}}(t) = \mathbf{g}(\mathbf{x}(t),\mathbf{y})$.
In a more specific form, my question is how to calculate this kind of integration:
$\int_0^T e^{-rt} f(x_1(t), x_2(t))dt$,
where $x_1'(t) = g_1(x_1(t))$, $x_2'(t) = g_2(x_2(t))$, and both $x_1$ and $x_2$ are scalars.