The discussion revolves around the integral of the sine function combined with logarithmic terms, specifically the expression $$\int_{0}^{2\pi}\sin\left({x\over 2}\right)\ln^2\left[\sin\left({x\over 4}\right)\sin\left({x\over 8}\right)\right]$$ and its proposed evaluation to $$\frac{27}{2}+\ln^2(2)+\ln(2)$$. The scope includes mathematical reasoning and problem-solving related to integrals.
There is no consensus on the validity or solvability of the integral presented. Some participants express skepticism about the complexity of the problem, while others are focused on the lack of solutions.
The discussion lacks specific mathematical steps or assumptions that might clarify the integral's evaluation. The nature of the integral and its proposed result remains unresolved.
Where are you getting these monstrosities from?Tony said:How to prove this integral,
$$\int_{0}^{2\pi}\sin\left({x\over 2}\right)\ln^2\left[\sin\left({x\over 4}\right)\sin\left({x\over 8}\right)\right]={27\over 2}+\ln^2(2)+\ln(2)$$
greg1313 said:Hi Tony and welcome to MHB! :D
Why are you posting these problems? Also, do you have the solutions?
I am considering moving them to the "Challenge Questions an Puzzles" forum and I can mark them as "Unsolved Challenges" if you do not have solutions.
Also, I encourage you to give more meaningful titles to your threads - I will be renaming several of them in the near future.
Good evening,
greg1313