Integral of (1/x) sin^101(x - 1/x) from 1/2 to 2

The same question, asked 2 times: 26 Apr 2025 S1*, 15 May 2026 S2. Every wording below, oldest first.

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Question 1
26 Apr 2025 S1
142 $\int_{\frac{1}{2}}^{2} \frac{1}{x}\csc^{101}\left( x-\frac{1}{x} \right)dx=$
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Question 2
15 May 2026 S2
83 $\int_{1/2}^{2}\dfrac{1}{x}\mathrm{\sin}^{101}\left(x - \dfrac{1}{x}\right) dx$
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