If \(\displaystyle \lim_{n \to \infty} \frac{(n+1)^{k-1}}{n^{k+1}} \left[(nk+1) + (nk+2) + \ldots + (nk+n)\right] = 33 \cdot \lim_{n \to \infty} n^\dfrac{1}{{k+1}} \left[1^k + 2^k + 3^k + \ldots + n^k\right] \), Then the integral value of \( k \) is equal to ________.
