Proving
$$\int_{0}^{\pi/2}(\log(\sin x))^2dx = \frac{1}{24}\cdot(\pi^3 + 12\pi(\log 2)^2)$$ without making use of gamma function,digamma function, hypergeometric function.
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Do you know complex analysis? – Potato Mar 13 '13 at 21:56
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2Alternatively: Do you know Fourier analysis? – Potato Mar 13 '13 at 21:58
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no and no. I prefer to avoid these ways. – Godly Light Mar 13 '13 at 22:15
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This is proven as $(9)$ here. – robjohn Mar 13 '13 at 22:15
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@robjohn: this is in my line. Thank you. – Godly Light Mar 13 '13 at 22:21
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I am closing this as a duplicate of this question since it is answered there. The two differ by a factor of $4$. – robjohn Mar 13 '13 at 22:26