Difference between revisions of "Relationship between integral of x*log(sin(x)), and Apéry's constant, pi, and logarithm"
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(Created page with "==Theorem== The following formula holds: $$\displaystyle\int_0^{\frac{\pi}{2}} x \log(\sin(x)) dx = \dfrac{7}{16}\zeta(3) - \dfrac{\pi^2}{8} \log(2).$$ ==Proof== ==Reference...") |
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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)) dx = \dfrac{7}{16}\zeta(3) - \dfrac{\pi^2}{8} \log(2),$$ |
| + | where $\pi$ denotes [[pi]], $\log$ denotes the [[logarithm]], $\sin$ denotes [[sine]], and $\zeta(3)$ denotes [[Apéry's constant]]. | ||
==Proof== | ==Proof== | ||
Revision as of 17:18, 24 June 2016
Theorem
The following formula holds: $$\displaystyle\int_0^{\frac{\pi}{2}} x \log(\sin(x)) dx = \dfrac{7}{16}\zeta(3) - \dfrac{\pi^2}{8} \log(2),$$ where $\pi$ denotes pi, $\log$ denotes the logarithm, $\sin$ denotes sine, and $\zeta(3)$ denotes Apéry's constant.
