RC step response equation derivation
A math error. 

If two functions are equal, it doesn't mean that ALL their integral functions are equal. You must add an integration constant (=an unknown number X which can be real or complex) to the equation, where two indefinite integrals are equal. That constant can be solved from boundary condition V=Vo when t=0.

So, continue from this:

$$\int{\frac{1}{V(t)-V_S}\frac{dV}{dt}dt}=X+\int{-\frac{1}{RC}dt}$$ 

Definite integrals declare different thing. They say that the integrals grow as much during the period from 0 to t and they are zero, if t=0.