Past exam of the short options paper shared by Prelims and Parts A and B of the physics course of the University of Oxford / 2017 / S25 / 1 / 2 / Solution

Multiplying $I^{\prime }-I=-\pi B$ by $e^{-\tau }$ gives $(I{e}^{-\tau }{)}^{\prime }=-\pi B{e}^{-\tau }$. Integrating to depth $T$ yields $I(0)=I(T)e^{-T}+\pi {\int }_0^{T}B(t)e^{-t}dt$. If the first term vanishes as $T\to \infty$,Each layer emits in proportion to $Bd\tau$ and is attenuated by its overlying optical depth. The normalized exponential weighting emphasizes optical depths of order unity, while retaining contributions from a range of heights.