How to do this interesting integration? 
$$\lim_{\Delta x\rightarrow0}\sum_{k=1}^{n-1}\int_{k+\Delta x}^{k+1-\Delta x}x^m dx$$

How to integrate the above integral?
Edit1: 
$$\lim_{\Delta x\rightarrow0}\int_{2-\Delta x}^{2+\Delta x}x^m dx$$
Does this intergral give $\space\space\space\space$ $2^m\space\space$  as the output?
Edit2:
Are my following steps correct?
$\lim_{\Delta x\rightarrow0}\sum_{k=1}^{n-1}\int_{k+\Delta x}^{k+1-\Delta x}x^m dx$ =
$\lim_{\Delta x\rightarrow0}\sum_{k=1}^{n-1}\int_{k+\Delta x}^{k+1-\Delta x}x^m dx$ $+$ $\lim_{\Delta x\rightarrow0}\sum_{k=1}^{n-1}\int_{k+1-\Delta x}^{k+1+\Delta x}x^m dx$  $-$ $\lim_{\Delta x\rightarrow0}\sum_{k=1}^{n-1}\int_{k+1-\Delta x}^{k+1 +\Delta x}x^m dx$ =
$\lim_{\Delta x\rightarrow0}\int_{1+\Delta x}^{n+\Delta x}x^m dx$ $-$ $\lim_{\Delta x\rightarrow0}\sum_{k=1}^{n-1}\int_{k+1-\Delta x}^{k+1 +\Delta x}x^m dx$ = 
$\lim_{\Delta x\rightarrow0}\int_{1}^{n}x^m dx$ $+$$\lim_{\Delta x\rightarrow0}\int_{n}^{n+\Delta x}x^m dx$ $-$  $\lim_{\Delta x\rightarrow0}\int_{1}^{1+\Delta x}x^m dx$ $-$
$\lim_{\Delta x\rightarrow0}\sum_{k=1}^{n-1}\int_{k+1-\Delta x}^{k+1 +\Delta x}x^m dx$ =
$\lim_{\Delta x\rightarrow0}\int_{1}^{n}x^m dx$ + 0 - 0 - 0 = $\lim_{\Delta x\rightarrow0}\int_{1}^{n}x^m dx$
= $\int_{1}^{n}x^m dx$

For the first question:
$$
\begin{align}
\lim_{\Delta x\to0}\,\left|\,\int_k^{k+1}x^m\,\mathrm{d}x-\int_{k+\Delta x}^{k+1-\Delta x}x^m\,\mathrm{d}x\,\right|
&=\lim_{\Delta x\to0}\,\left|\,\int_k^{k+\Delta x}x^m\,\mathrm{d}x+\int_{k+1-\Delta x}^{k+1}x^m\,\mathrm{d}x\,\right|\\
&\le\lim_{\Delta x\to0}2\Delta x(k+1)^m\\
&=0
\end{align}
$$
we get
$$
\begin{align}
\lim_{\Delta x\to0}\sum_{k=1}^{n-1}\int_{k+\Delta x}^{k+1-\Delta x}x^m\,\mathrm{d}x
&=\sum_{k=1}^{n-1}\int_k^{k+1}x^m\,\mathrm{d}x\\
&=\int_1^nx^m\,\mathrm{d}x
\end{align}
$$
For the Edit:
For $\Delta x<1$,
$$
\left|\,\int_{2-\Delta x}^{2+\Delta x}x^m\,\mathrm{d}x\,\right|\le2\cdot3^m\Delta x
$$
Therefore,
$$
\lim_{\Delta x\to0}\,\int_{2-\Delta x}^{2+\Delta x}x^m\,\mathrm{d}x=0
$$
For your steps:
If you would give the justification for each step, it would help us in commenting on what is correct and what might be wrong and help you in seeing what is right.
