Integrating factor: renaming variable I'm having trouble following some math in a manuscript. The author is using an integrating factor to begin to solve a differential equation:
$$
dX(t)=[\mu(t)+b(t) X(t)]dt + \sigma(t)dB(t)
$$
by introducing the integrating factor 
$$
e^{-\int_{}^{t} b(\tau) d\tau}
$$
$b()$ is not the derivative of $B()$. The given solution is:
$$
X(t)=\int_{0}^{t} e^{\int_{\tau}^{t} b(s) ds}\mu(\tau)d\tau+\int_{0}^{t} e^{\int_{\tau}^{t} b(s) ds}\sigma(\tau)dB(\tau)
$$
My questions is, why is this variable renaming allowed? I find something similar in the wikipedia page for integrating factors, but it's not clear to me why it's allowed (required?), or what the motivation is.

We use this fact to simplify our expression to $$ \frac{d}{dx}(y
 e^{\int_{s_0}^{x} P(s) ds}) = Q(x) e^{\int_{s_0}^{x} P(s) ds}  $$ We
  then integrate both sides with respect to x, firstly by renaming x to
  t, obtaining $$ y e^{\int_{s_0}^{x} P(s) ds} = \int_{t_0}^{x} Q(t)
 e^{\int_{s_0}^{t} P(s) ds} dt + C $$

 A: The integrating factor can be chosen to be any one of infinitely many functions ($a$ can be anything): $$f_a(t)=\exp\left({-\int_a^tb(\tau)\,\mathrm d\tau}\right)$$
If you're asking why I'm using $\tau$ there, it doesn't matter what I use there because it's a "dummy variable". In fact, another notation would allow me to write more simply:  $$f_a(t)=\exp\left({-\int_a^tb}\right)$$ However, if I used $t$ there and wrote something like $-\int_a^tb(t)\,\mathrm dt$, I might get confused because the $t$ on the inside that just tells me what function related to $b$ I'm integrating is a completely different thing than the $t$ on the outside which is the input of $f_a$.
Now, in the manuscript, the author does a nice little trick since a definite integral changes sign when you swap the limits of integration, so that $-\int_a^tb(\tau)\,\mathrm d\tau=\int_t^ab(\tau)\,\mathrm d\tau$.
If we want to substitute this into an even bigger expression, and $t$ is reserved we should use a third variable to avoid confusion, and write something like
$$\int_{0}^t\left(\exp\left(\int_s^ab(\tau)\,\mathrm d\tau\right)\right)\,\mathrm ds$$
However, for stylistic reasons, $\tau$ is traditionally kept closer to $t$/the outside, so we may write the following instead:
$$\int_{0}^t\left(\exp\left(\int_\tau^ab(s)\,\mathrm ds\right)\right)\,\mathrm d\tau$$

Now, the question is, why does your question have something more like: $$\int_{0}^t\left(\exp\left(\int_\tau^tb(s)\,\mathrm ds\right)\right)\,\mathrm d\tau$$
with $a$ changed into $t$?
It's because the solving of the differential equation goes something like:
$$\exp\left({-\int_a^tb(s)\,\mathrm ds}\right)X'(t)=\exp\left({-\int_a^tb(s)\,\mathrm ds}\right)b(t)X(t)+\exp\left({-\int_a^tb(s)\,\mathrm ds}\right)\text{stuff}$$
$$\Rightarrow\exp\left({-\int_a^tb(s)\,\mathrm ds}\right)X'(t)-\exp\left({-\int_a^tb(s)\,\mathrm ds}\right)b(t)X(t)=\exp\left({-\int_a^tb(s)\,\mathrm ds}\right)\text{stuff}$$
$$\Rightarrow\exp\left({-\int_a^tb(s)\,\mathrm ds}\right)X(t)+C_0=\int_0^t\exp\left({\int_\tau^a b(s)\,\mathrm ds}\right)\text{stuff}\,\mathrm d \tau$$
$$\Rightarrow X(t)+C_0=\int_0^t\exp\left({\int_a^tb(s)\,\mathrm ds}\right)\exp\left({\int_\tau^a b(s)\,\mathrm ds}\right)\text{stuff}\,\mathrm d \tau$$
$$\Rightarrow X(t)+C_0=\int_0^t\exp\left({\int_a^tb(s)\,\mathrm ds}+{\int_\tau^a b(s)\,\mathrm ds}\right)\text{stuff}\,\mathrm d \tau$$
$$\Rightarrow X(t)+C_0=\int_0^t\exp\left({\int_\tau^tb(s)\,\mathrm ds}\right)\text{stuff}\,\mathrm d \tau$$
