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The notion of the Hadamard product is extended to evaluate an extensive number of trigonometric integrals in terms of sums. These sums are taken over index sets defined by a Diophantine equation which can be simplified in certain circumstances. The results obtained include generalizations of integrals defining sums of products of Bessel functions and integrals of powers of cosines evaluated in terms of sums of products of binomial coefficients. Generating functions for special polynomials aredoi:10.1216/rmjm/1181069890 fatcat:efu5pdyarvf7xnwykvpwv7zaq4