@mmcdara Thank you for your response. This is making me realize that I am really out of my depth here haha. Yes, the name PDF means probability density function in this case. The sampling routine in my original post transforms two random numbers x and y into a sample of DVN, the distribution of visible normals. Ve then gets reflected along this sampled normal, and PDF is the probability density function for the reflected vectors. You are right though, I realized an error in my earlier post: I am implicitly assuming that Ve is in the upper hemisphere, so VeZ is in [0, 1], not [-1, 1]. I'll check if that changes anything about your comments.

@dharr alphaX and alphaY are in (0, 1], VeX, VeY and VeZ are in [-1, 1] (Ve is a unit vector), and x and y are in [0, 1], and [xInf, xSup] and [yInf, ySup] are subsets on that.

@dharr I double checked the source (Eric Heitz, Sampling the GGX Distribution of Visible Normals, Journal of Computer Graphics Techniques (JCGT), vol. 7, no. 4, 1–13, 2018) and it's correct. I'm not asking about the correctness of my code, though. I only provided it so that maybe people can test it themselves and help more easily with my problem.

@salim-barzani I can share the file once I'm home, but the code in my post is literally all there is 🙈

Also, sorry, I'm not sure I'm quite following you. I didn't use Ai for anything, not my code and not my post. Like I said, I'm very new to Maple. I need the integral for my code and it's too long to do by hand, so if my code looks a bit clunky, it's literally because I've been using Maple for two days.

Edit: If you think I used Ai because of my phrasing "the code that is given", I just mean that up until that, everything that I wrote is correct and not up for debate, not that Ai or anyone else wrote it 😅