Integrating a function of φ and cosθ over the entire solid angle in spherical coordinates.

Question

I found this integral in a book:

link here

The integrated function is a phase function or just a probability density function.

I don't understand why instead of dω' = sinθ dφ dθ in this case dω' = dφ dcosθ.

Answer 1

Hint: $$ \frac{d \cos\theta}{d\theta} = -\sin \theta, $$ and I suppose the minus sign is absorbed in $p(\cos \theta)$.

If the above is incorrect, please comment more details pertaining to the question.

Thanks!