I am studying the book Gamma by Julian Havil and there are two equations involving Bernoulli Numbers stated without proofs: $$\sum_{k=1}^{n}k^m = \frac1{m+1} \sum_{k=1}^{m} {{m+1}\choose k-1} B_{k-1}n^{m-k+1} $$ and $$\sum f(k) = \int f(x)dx + \frac12 (f(1) + f(n)) + \sum \dfrac{B_{2k}}{(2k)!} (f^{2k-1}(n)+f^{2k-1}(1))+R.$$
Is there any good self-learnable explanatory books includes proofs of those two mentioned equations and perhaps more alike equations?