MHB Simple closed form for integral

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The discussion revolves around finding a closed form for the integral involving trigonometric and logarithmic functions. The integral in question is $$\int_{0}^{1}t\cos(2t\pi)\tan(t\pi)\ln[\sin(t\pi)]\mathrm dt$$ and is proposed to equal $$\frac{1}{\pi}\cdot\frac{\ln 2}{2}(1-\ln 2)$$. Participants express a desire for hints to approach the solution, indicating a lack of progress over time. The request for assistance remains unanswered after multiple follow-ups. The conversation highlights the challenge of solving complex integrals in mathematical analysis.
Tony1
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How may we go about to show that,

$$\int_{0}^{1}t\cos(2t\pi)\tan(t\pi)\ln[\sin(t\pi)]\mathrm dt=\color{green}{1\over \pi}\cdot\color{blue}{{\ln 2\over 2}(1-\ln 2)}$$
 
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A hint is requested ... (Blush)
 
lfdahl said:
A hint is requested ... (Blush)
can I get the hint?
 
64 days after my 1st request:

A hint is still requested ... (Wave)
 

Thread 'Area of Overlapping Squares'

Here is a little puzzle from the book 100 Geometric Games by Pierre Berloquin. The side of a small square is one meter long and the side of a larger square one and a half meters long. One vertex of the large square is at the center of the small square. The side of the large square cuts two sides of the small square into one- third parts and two-thirds parts. What is the area where the squares overlap?
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