Integrating a differential and notation of limits of integration

Question

I am studying Thermodynamics and in class I came up with integrating the differential of internal energy:$$\int^2_1dU=U_2-U_1$$ But I don't understand why someone couldn't just write $$\int^2_1dU=U|^2_1=2-1=1$$ So how can we know if the integral mean the difference between two different function states?

Answer 1

It's an abuse of notation. If $U$ is a function of $t$, say, then $dU=U'(t)\,dt$, the initial and final states are $U_1=U(1)$, $U_2=U(2)$. So what the integral is trying to mean is $$ \int_1^2\,dU=\int_1^2\,U'(t)\,dt=\left.U(t)\right|_1^2=U(2)-U(1). $$