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For a special value of a parameter, the identity reduces to a definite integral with a very simple symbolic value; but, quite surprisingly, all the consulted tables of integrals, and computer algebra systems, do not seem aware of this result. Here I show that this result is a special case ($n=0$ and $z=1$) of the following identity (established by elementary methods): $$ I_n(z)\\equiv\\int_0^1{{\\rm K}(k) k\\over (z+k^2)^{n+3/2}}dk = {(-2)^n\\over (2n+1)!!} {d^n\\over dz^n} {{\\r"},"verification_status":{"content_addressed":true,"pith_receipt":true,"author_attested":false,"weak_author_claims":0,"strong_author_claims":0,"externally_anchored":false,"storage_verified":false,"citation_signatures":0,"replication_records":0,"graph_snapshot":true,"references_resolved":false,"formal_links_present":false},"canonical_record":{"source":{"id":"1911.10480","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CA","submitted_at":"2019-11-24T09:15:23Z","cross_cats_sorted":[],"title_canon_sha256":"19b1a4b9302e3d3521766c2c58c42a0561c7e861693b2d56175798051d28e8fd","abstract_canon_sha256":"752e7968a8612cc4e6469763aefa6b725c1ee67dc5dd4c0c348ef9ffd277ba6a"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-07-05T00:22:43.672727Z","signature_b64":"6wrAXyht0TRoochs9RJqj1dLf+OE8c74VX1LjjiRIRSaLMqovfSNeKDtKsd1J9hr67/SdtxhMtLWah3a6vmdCg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"186e5f82e2bf2cdabbed0a8204b174c0e2774f541eae3aacc286eee10a666650","last_reissued_at":"2026-07-05T00:22:43.672340Z","signature_status":"signed_v1","first_computed_at":"2026-07-05T00:22:43.672340Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"On a family of curious integrals suggested by Stellar Dynamics","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CA","authors_text":"Astronomy, Italy), Luca Ciotti (Dept. of Physics, University of Bologna","submitted_at":"2019-11-24T09:15:23Z","abstract_excerpt":"While investigating the properties of a galaxy model used in Stellar Dynamics, a curious integral identity was discovered. 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