Consider an ODE of the form:
$\displaystyle \frac{dy}{dx}+P(x)y=Q(x)$
Now, if we define an "integrating factor" as follows:
$\displaystyle \mu(x)=\exp\left(\int P(x)\,dx\right)$
Then it follows that:
$\displaystyle \frac{d\mu}{dx}=P(x)\mu(x)$
And so, multiplying through the ODE by this factor, there results:
$\displaystyle \mu(x)\frac{dy}{dx}+P(x)\mu(x)y=\mu(x)Q(x)$
And we may then write:
$\displaystyle \mu(x)\frac{dy}{dx}+\frac{d\mu}{dx}y=\mu(x)Q(x)$
At this point we should observe that:
$\displaystyle \frac{d}{dx}\left(\mu(x)\cdot y\right)=\mu(x)\frac{dy}{dx}+\frac{d\mu}{dx}y$
And so the ODE may be written:
$\displaystyle \frac{d}{dx}\left(\mu(x)\cdot y\right)=\mu(x)Q(x)$