Difference between revisions of "Relationship between integral of x*log(sin(x)), and Apéry's constant, pi, and logarithm"

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==Theorem==
 
==Theorem==
 
The following formula holds:
 
The following formula holds:
$$\displaystyle\int_0^{\frac{\pi}{2}} x \log(\sin(x)) dx = \dfrac{7}{16}\zeta(3) - \dfrac{\pi^2}{8} \log(2),$$
+
$$\displaystyle\int_0^{\frac{\pi}{2}} x \log(\sin(x)) \mathrm{d}x = \dfrac{7}{16}\zeta(3) - \dfrac{\pi^2}{8} \log(2),$$
 
where $\pi$ denotes [[pi]], $\log$ denotes the [[logarithm]], $\sin$ denotes [[sine]], and  $\zeta(3)$ denotes [[Apéry's constant]].
 
where $\pi$ denotes [[pi]], $\log$ denotes the [[logarithm]], $\sin$ denotes [[sine]], and  $\zeta(3)$ denotes [[Apéry's constant]].
  

Latest revision as of 17:18, 24 June 2016

Theorem

The following formula holds: $$\displaystyle\int_0^{\frac{\pi}{2}} x \log(\sin(x)) \mathrm{d}x = \dfrac{7}{16}\zeta(3) - \dfrac{\pi^2}{8} \log(2),$$ where $\pi$ denotes pi, $\log$ denotes the logarithm, $\sin$ denotes sine, and $\zeta(3)$ denotes Apéry's constant.

Proof

References