  We give a simple proof for the reciprocity formulas of character Dedekind
sums associated with two primitive characters, whose modulus need not to be
same, by utilizing the character analogue of the Euler-MacLaurin summation
formula. Moreover, we extend known results on the integral of products of
Bernoulli polynomials by considering the integral \[
\int\limits_{0}^{x}B_{n_{1}}(b_{1}z+y_{1})... B_{n_{r}}(b_{r}z+y_{r}) dz, \]
where $b_{l}$ $(b_{l}\neq 0)$ and $y_{l}$ $(1\leq l\leq r)$ are real numbers.
As a consequence of this integral we establish a connection between the
reciprocity relations of sums of products of Bernoulli polynomials and of the
Dedekind sums.
