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I want to proof the series of kummer. Therefore I want to show $$2 \cdot \int_0^1 \log\big(\Gamma(x)\big) \cdot \sin(2 \pi n x) dx = \frac{\gamma + \log(2 \pi) + \log(n)}{n \pi},$$ where $\gamma$ is the Euler-Mascheroni-constant.

Any help would be appreciated. Thanks in advance.

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