In my textbook of physical chemistry I encountered strange derivation: by integrating
$$dU=TdS-pdV+\sum_{i=0}^n \mu_i dn_i $$ we get $$U=TS-pV+\sum_{i=0}^n \mu_i n_i . $$ I tried to understand this. dU is total differential, that means that I should solve it like path integral, and I can chose any path. I know that from these it possible to show that integral of total differential is sum of definite integrals. I am not sure what start and end points I should chose. I assume that there are some hidden assumptions in text, that should be self-evident. $$U=\int_0^S T(S,V,A, n_1, n_2...)dS-\int_0^p p(S,V,A, n_1, n_2...)dV+ \int_0^\mu \mu_i(S,V,A, n_1, n_2...) dn_i +C.$$ Please show step by step solution and book where its written, preferably older book.
The idea is the following. We assume that $U(S,V,n)$ is a homogeneous function of order 1 that is $U(\lambda S, \lambda V, \lambda n)=\lambda U(S,V,n)$ (physically it means that if we increas the system $\lambda$ times then the energy increase $\lambda$ times, in other words $U$ is extensive). Then we can use Euler's Homogeneous Function Theorem (https://mathworld.wolfram.com/EulersHomogeneousFunctionTheorem.html) so we get \begin{equation} U(S,V,n ) =\Big(\frac{\partial U}{\partial S}\Big) S +\Big(\frac{\partial U}{\partial V}\Big) V+ \Big(\frac{\partial U}{\partial n}\Big) n \end{equation} Where we know the partial derivatives from the equation $dU=TdS-pdV+\mu n$ (for example $\frac{\partial U}{\partial S}=T $ and so on..) Finally we get \begin{equation} U(S,V,n ) =T S -pV+\mu n \end{equation}