How do you prove the result?
$$\int_{0}^{\infty} \frac{\sin(\pi x)}{x(1-x^2)}dx = \pi $$
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1Does this answer your question? https://math.stackexchange.com/questions/114086/using-residue-theory-to-evaluate-int-0-infty-frac-sin-pi-xx1-x2 – Sumanta Sep 23 '20 at 11:52
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Indeed it does provide a prove but uses residue theorem. I was interested in a more classical approach. I got as far as splitting into partial fractions but the resulting integrals (all of similar form) are difficult themselves. – Callie12 Sep 23 '20 at 11:57
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Apologies, perhaps I should have stated 'by elementary' methods. – Callie12 Sep 23 '20 at 12:11
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@Callie12 By partial fractions, you may get $I = 2\int_0^\infty \frac{\sin \pi x}{x} \mathrm{d}x$, right? – River Li Sep 23 '20 at 14:59