The average power at output when additive Gaussian white noise is present is given by the formula below:
$$P_0 = \int\limits_{-B}^{B} \frac N2\ df ,\quad \text{with $B$ being the bandwidth.}$$
My question is when the bandwidth is in terms of $\textrm{rad/s}$, what would be the formula? The one stated divided by $2\pi$ or another formula?
Just to be clear about the units:
\begin{align} \Omega &= 2\pi F, \quad \text{with $\Omega$ in } \rm\underline{radians/sec}\text{ and $F$ in } \rm\underline{cycles/sec} \text{ or }\rm\underline{Hz}\tag{1}\\ \omega &= 2\pi f, \quad \text{with $\omega$ in } \rm\underline{radians/samples}\text{ and $f$ in } \rm\underline{cycle/sample}\tag{2}\\ \omega&=\Omega T_s\quad \text{and}\quad f=F/F_s\tag{3} \end{align} Read @Dilip Sarwate's comments; what should happen there is a change of variables. \begin{align} \omega &= 2\pi f\Longrightarrow d\omega = 2\pi df\iff df=\frac{d\omega}{2\pi}\tag{4}\\ f&=\pm B \Longrightarrow \omega= \pm 2\pi B\tag{5}\\ \end{align} Using $(4)$ and $(5)$ you then have the following results: $$ P_0=\int_{-B}^{B} \frac N2 df = NB \Longrightarrow P_0= \int_{-2\pi B}^{2\pi B} \frac N2 \cdot \frac{1}{2\pi}d\omega=\frac{1}{2\pi}\int_{-2\pi B}^{2\pi B} \frac N2 d\omega = NB\tag{6}\\ $$
You are asking two questions, Let me answer the question concerning the power of your noise:
you integrate from $-B$ to $B$. If we do integrate the term we get:
$$ P_o = \int_{-B}^B \frac{N}{2} df = \left. \left[ \frac{N}{2}f \right] \right\rvert_{-B}^{B} = \frac{N}{2} \left[ B + B \right] = NB $$
So your noise scales linear with the bandwidth. Assuming our noise term is defined as $$ [P_{gauss}] = \left[\frac{V^2}{Hz}\right] $$ we need to use the frequency and not the angular frequency for the bandwidth. You can obtain it from the following formula: $$ \omega = 2\pi f $$ with the units $$ \left[\frac{rad}{s}\right] = 2\pi [Hz] = \underbrace{2\pi}_{"rad"} \left[\frac{1}{s}\right] $$
Therefore, if you have$P_{gauss}$ defined in the unit $Hz$ you need to correct for $2\pi$ like so $$ P_{gauss, \omega} = \frac{NB}{2\pi} $$
Please keep in mind that $2\pi$ does not have unitx. The term $rad$ means radians and is only added to be able to distinguish between the frequency in Hz and the angular frequency. In signal processing we often deal with sines, cosines and thelike which have a periodicity of $2\pi$, so $\omega$ is often used to simplify the calculations.
