\frac{1}{\pi}\int_{-\pi}^{\pi}x/\pi \sin(nx)\, dx - Wolfram|Alpha

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Definite integral:

link to /input/?i=%28-2+n+%CF%80+cos%28n+%CF%80%29%2B2+sin%28n+%CF%80%29%29%2F%28n%5E2+%CF%80%5E2%29&lk=1&a=ClashPrefs_*Math-
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Series expansion of the integral at n=0:

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Indefinite integral:

link to /input/?i=%28-%28x+cos%28n+x%29%29%2Fn%2B%28sin%28n+x%29%29%2Fn%5E2%29%2F%CF%80%5E2&lk=1&a=ClashPrefs_*Math-
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