Animated Solution for Mathematics - Definite Integration: If ${\lim }_{n\to \infty }\frac{(n+1{)}^{k-1}}{n^{k+1}}[(nk+1)+(nk+2)+\cdots +(nk+n)]=33\cdot {\lim }_{n\to \infty }\frac{1}{n^{k+1}}[1^k+2^k+3^k+\cdots +n^k]$, then the integral value of $k$ is equal to ______.

We are tasked with solving a limit of a summation, a classic challenge that requires breaking down complex expressions into manageable components. We begin by examining the Left Hand Side (LHS) of the equation:

The terms within the summation are $(nk+1),(nk+2),\dots ,(nk+n)$, which form an arithmetic progression. We can decompose this sum into two distinct parts:

The first part is simply adding $nk$ to itself $n$ times, yielding $n^{2}k$. The second part is the sum of the first $n$ natural numbers, which is $\frac{n(n+1)}{2}$. Thus, the bracketed sum simplifies to:

Now, we incorporate the limit:

To simplify, we factor $n^2$ out of the bracket:

Simultaneously, we rewrite the coefficient as:

As $n\to \infty$, the $n^2$ terms cancel out, and the terms involving $\frac{1}{n}$ vanish. We are left with $1^{k-1}\cdot (k+\frac{1}{2})$, which simplifies to:

Next, we address the Right Hand Side (RHS):

We rewrite this expression to fit the definition of a Riemann sum:

This expression transforms directly into a definite integral:

We equate the simplified LHS and RHS to form an algebraic equation:

Multiplying both sides by $2(k+1)$ to clear the denominators, we obtain:

Expanding this results in the quadratic equation:

Factoring the quadratic, we find:

This yields two potential values: $k=5$ and $k=-\frac{13}{2}$. Since the problem requires an integer solution, we discard the fractional value. The final answer is $k=5$.