The formula for the sums of powers of positive integers, given by Faulhaber
in 1631, is proven by using trigonometric identities and some properties of
the Bernoulli polynomials. Using trigonometric functions identities and
generating functions for some well-known special numbers and polynomials,
many novel formulas and relations including alternating sums of powers of
positive integers, the Bernoulli polynomials and numbers, the Euler
polynomials and numbers, the Fubini numbers, the Stirling numbers, the
tangent numbers are also given. Moreover, by applying the Riemann integral
and p-adic integrals involving the fermionic p-adic integral and the
Volkenborn integral, some new identities and combinatorial sums related to
the aforementioned numbers and polynomials are derived. Furthermore, we
serve up some revealing and historical remarks and observations on the
results of this paper.