I'm trying to solve problems which is about the application of indefinite integral. I do quite understand how to solve it but at some point I'm curious where does the $ln(C)$ came from.
On the first question I'm expecting the when I simplify $$\int \frac{dq}{q}=\int kdt$$. It will just become $$q=kt+C$$ but on the solution it has $ln(C)$. Same with the second question but on the second question I would like to ask also how it becomes $-kt$ instead of only $kt$.
The constant of integration $+C$ is just letting you know "and adding any real constant to this solution also gives me a solution". So, $+2C$ would do the same job: you get the same solution space. $+\log(C)$ does the same thing, since $\log$ has range the whole of $\mathbb{R}$. (In this type of problem, experience will show that adding $\log(C)$ is a good thing to do, but if you hadn't seen this type of equation before, $+C$ is fine).
As for your second question, try differentiating the left hand side of the solution: you will find that a minus sign turns up.