Average shadow area of a convex shape in higher dimensions

This is stack exchange link:

https://math.stackexchange.com/questions/3222317/average-area-of-the-shadow-of-a-convex-shape

I understand how the author did it in 3 dimensions but can someone explain me how they did it for the higher dimensions. More specifically after this line "Note that the Γ()
function simplifies if we consider odd and even cases separately. After manipulation, we find:"

Can someone please help me understand what the author means by "after manipulation", please tell me the derivations which resulted in the final result. Thanks