# How is the beam transmittance calculated in PBRT V3?

In pbrt v3, the book gives this description of beam transmittance, but I don't know how to solve the differential equation like it says to get Tr , can someone please tell me how to solve the differential equation? Thanks a lot.

To review that standard derivation: suppose we want a function $$y(x)$$ obeying the differential equation $$\mathrm{d}y/\mathrm{d}x = -ky$$, for some constant $$k$$. Then we can solve the equation as follows: \begin{aligned} \mathrm{d}y &= -ky \, \mathrm{d}x \\ \frac{\mathrm{d}y}{y} &= -k \, \mathrm{d}x \\ \int \frac{\mathrm{d}y}{y} &= -\int k \, \mathrm{d}x \\ \ln y &= -kx \\ y &= e^{-kx} \end{aligned} (there should actually be some constants of integration in there, but I left them out since they're not important for this answer.)
Now, suppose we generalize and make the constant $$k$$ into a function $$k(x)$$. Then we can repeat this derivation, but we will not be able to do the integral on the right side, since $$k(x)$$ is unspecified. The result will then be: $$y = e^{-\int k(x) \, \mathrm{d}x}$$
The derivation for the transmittance along a ray is just the same, but with some variables renamed: $$y$$ becomes $$L$$, $$x$$ is now the parameter $$t$$ along the ray, and $$k(x)$$ is called $$\sigma_\mathrm{t}(\mathrm{p})$$.