First of a series of trig integrals: Show that
\[\lim_{n\to\infty}\int_{\pi/4}^{3\pi/4}\sum_{k=1}^n\sin(2k-1)x\,dx=\ln\left(\cot\frac{\pi}{8}\right)\]

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The next one looks a bit tougher, I'll post it once this one's resolved.

(I suppose that qualifies as a hint)

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