Sorry if this is a really basic question but I can't seem to get my head around the steps involved in this integration at all. My equation to be integrated is as follows:
${ds \over s}=\mu dt$
Integrating between $t=0$ and $t=T$ supposedly yields the following:
$s(T) = s(0)e^{\mu T}$
What are the intermediate steps to get here?
Try taking the indefinite integral of both sides instead: \begin{align*} \frac{ds}{s} &= \mu \, dt \\ \int \frac{1}{s} \, ds &= \int \mu \, dt \\ \ln|s| &= \mu t + C &\text{for some constant $C \in \mathbb R$}\\ e^{\ln|s|} &= e^{\mu t + C}\\ |s| &= e^{\mu t} \cdot e^C \\ s &= \underbrace{(\pm e^C)}_{A}e^{\mu t} \\ s(t) &= Ae^{\mu t} &\text{for some constant $A \in \mathbb R$}\\ \end{align*} To solve for $A$, notice that if we substitute $t = 0$, then we get: $$ s(0) = Ae^{\mu \cdot 0} = A \cdot 1 = A $$ Hence, by taking $t = T$, we conclude that: $$ s(T) = s(0)e^{\mu T} $$