I am looking at a derivation of the solution to the Heat Equation for my partial differential equations course, but I am stuck on one part of the derivation and I want to understand everything that is happening. Below is an excerpt from the notes:
Recalling that $Q(x, t) = g(p)$, where $p = \frac{x}{\sqrt{4kt}}$, we obtain the following explicit formula for $Q$ $$Q(x,t) = c_1\int_0^{\frac{x}{\sqrt{4kt}}} e^{-p^2}dp + c_2 $$
Now, my question is why we have chosen the bounds of integration to be $(0,\frac{x}{\sqrt{4kt}})$. This seems rather arbitrary. Could we have chosen something else as the bounds? This method seems to be standard in all sources I have checked so I believe there must be a concrete reason. I am super stuck on this so any help would be greatly appreciated. Here is a screenshot of the notes if anyone needs more reference.
