# 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?

$\lim_{\Delta x\rightarrow0}\sum_{k=1}^{n-1}\int_{k+\Delta x}^{k+1-\Delta x}x^m dx$ =

$\lim_{\Delta x\rightarrow0}\sum_{k=1}^{n-1}\int_{k+\Delta x}^{k+1-\Delta x}x^m dx$ $+$ $\lim_{\Delta x\rightarrow0}\sum_{k=1}^{n-1}\int_{k+1-\Delta x}^{k+1+\Delta x}x^m dx$ $-$ $\lim_{\Delta x\rightarrow0}\sum_{k=1}^{n-1}\int_{k+1-\Delta x}^{k+1 +\Delta x}x^m dx$ =

$\lim_{\Delta x\rightarrow0}\int_{1+\Delta x}^{n+\Delta x}x^m dx$ $-$ $\lim_{\Delta x\rightarrow0}\sum_{k=1}^{n-1}\int_{k+1-\Delta x}^{k+1 +\Delta x}x^m dx$ =

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