Situations in which to express integration constant as its natural log What is the complete reasoning behind constants of integration, specifically in the case of the natural log? Given $\int\left(\frac{1}{250-x}\right)dx=\int(dt)$ (1), I obtain: $\ln\left(\frac{1}{250-x}\right)=t+C$ My textbook gives the solution for this problem as: $\ln\left(\frac{1}{250-x}\right)=t+\ln(C)$ However, for another basic problem, we do not end up taking the natural log of the constant of integration: $\int\left(\frac{1}{x(1-x)}\right)dx=\int(dt)$ (2) The text finds: $\ln\left(\frac{x}{x-1}\right)=t+C$ Why is the constant of integration expressed as its natural log in the case of 1, but not in the case of 2?

It depends upon the context of the problem. In your first example it appears that perhaps one intends to solve for $x$ but in the second example it appears than one does not intend to solve for $x$. In the first example one immediately obtains \begin{equation} \frac{1}{250-x}=Ce^t \end{equation} by having expressed the constant in the form $\ln C$. If one wished to solve the second example for $x$ it would be better to use $\ln C$ there as well.