Proofs of McIntosh's Conjecture on Franel Integrals and Two Generalizations

We provide a proof of a conjecture made by Richard McIntosh in 1996 on the values of the Franel integrals, $${\int }_0^1((ax))((bx))((cx))((ex))dx,$$ where $((x))$ is the first periodic Bernoulli function. Secondly, we extend our ideas to prove a similar theorem for $${\int }_0^1((a_{1}x))((a_{2}x))\cdots ((a_{n}x))dx\mathrm{.}$$ Lastly, we prove a further generalization in which $((x))$ is replaced by any particular Bernoulli function with odd index.