Say you had the differential equation
$$\frac{\mathrm{d}x}{\mathrm{d}t} = f(x,t)$$
Separating variables, you get $$\mathrm{d}x = f(x,t)\,\mathrm{d}t$$
From there, when I try to solve for $x$, I integrate both sides
$$x = \int f(x,t)\,\mathrm{d}t$$
Without an explicit function, how do you continue to solve this?
I've seen differentiation under the integral sign, which says
$$\frac{\mathrm{d}}{\mathrm{d}t}\int f(x,t)\,\mathrm{d}t = \int\frac{\partial}{\partial t}f(x,t)\,\mathrm{d}t$$
but would that mean I'd need to differentiate both sides, and I'd end up with $\frac{\mathrm{d}x}{\mathrm{d}t}$ on the left side again?
There's no nice way to write this, what you have is the best way to write this. To see why it's so hopeless, I offer a much easier problem. Given a function $f(x,t)$, can you write a general form for
$$\frac{\partial}{\partial t}f(x,t)$$
If you can't even find a formula for this, integration, which is much harder, is even more hopeless.