= Solution
Multiplying $I'-I=-\pi B$ by $e^{-\tau}$ gives $(I e^{-\tau})'=-\pi B e^{-\tau}$. Integrating to depth $T$ yields $I(0)=I(T)e^{-T}+\pi\int_0^TB(t)e^{-t}dt$. If the first term vanishes as $T\to\infty$,
$$
\boxed{I_0=\pi\int_0^\infty B(\tau)e^{-\tau}d\tau.}
$$
Each layer emits in proportion to $B\,d\tau$ and is attenuated by its overlying <optical depth>. The normalized exponential weighting emphasizes <Optical depth>[optical depths] of order unity, while retaining contributions from a range of heights.
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