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Question Number 142424 by Mathspace last updated on 31/May/21

2)calculate Σ_(k=1) ^(n−1)  sin(((kπ)/n))   (n>2)  1) use Rieman sum to prove  that ∫_0 ^π log(sinx)dx=−πlog2

$$\left.\mathrm{2}\right){calculate}\:\sum_{{k}=\mathrm{1}} ^{{n}−\mathrm{1}} \:{sin}\left(\frac{{k}\pi}{{n}}\right)\:\:\:\left({n}>\mathrm{2}\right) \\ $$ $$\left.\mathrm{1}\right)\:{use}\:{Rieman}\:{sum}\:{to}\:{prove} \\ $$ $${that}\:\int_{\mathrm{0}} ^{\pi} {log}\left({sinx}\right){dx}=−\pi{log}\mathrm{2} \\ $$

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