The integral $$\int_{0}^{2\pi}\sin\left({x\over 2}\right)\ln^2\left[\sin\left({x\over 4}\right)\sin\left({x\over 8}\right)\right]$$ is proposed to equal $$\frac{27}{2}+\ln^2(2)+\ln(2)$$. Participants in the discussion express curiosity about the origin of this integral and the absence of provided solutions. The discussion suggests a potential relocation of the problem to a "Challenge Questions and Puzzles" forum if no solutions are available.
PREREQUISITESMathematicians, students of calculus, and anyone interested in solving complex integrals or engaging in mathematical problem-solving discussions.
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