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# integration of polynomial

###### Theorem.

For all nonnegative integers $n$,

$\int x^{n}\,dx=\frac{1}{n+1}x^{{n+1}}+C.$ |

###### Proof.

It will first be proven that, for any nonnegative integer $n$ and any $a\in\mathbb{R}$,

$\int\limits_{0}^{a}x^{n}\,dx=\frac{1}{n+1}a^{{n+1}}.$ |

If $a=0$, the above statement is obvious. If $a>0$, the following computation uses the right hand rule for computing the integral; if $a<0$, the following computation uses the left hand rule for computing the integral:

$\displaystyle\int\limits_{0}^{a}x^{n}\,dx$ | $\displaystyle=\lim_{{t\to\infty}}\sum_{{k=1}}^{t}\left(\frac{ak}{t}\right)^{n}% \left(\frac{a}{t}\right)$ |
---|---|

$\displaystyle=a^{{n+1}}\lim_{{t\to\infty}}\frac{1}{t^{{n+1}}}\sum_{{k=1}}^{t}k% ^{n}$ | |

$\displaystyle=a^{{n+1}}\lim_{{t\to\infty}}\frac{1}{t^{{n+1}}}\sum_{{l=1}}^{{n+% 1}}{n+1\choose r}\frac{B_{{n+1-l}}}{n+1}(t+1)^{l}$ by this theorem, | |

$\displaystyle=a^{{n+1}}\lim_{{t\to\infty}}\frac{1}{t^{{n+1}}}{n+1\choose n+1}% \frac{B_{{n+1-(n+1)}}}{n+1}(t+1)^{{n+1}}$ | |

$\displaystyle=\frac{B_{0}}{n+1}a^{{n+1}}\lim_{{t\to\infty}}\left(\frac{t+1}{t}% \right)^{{n+1}}$ | |

$\displaystyle=\frac{1}{n+1}a^{{n+1}}$ |

Thus, if $a,b\in\mathbb{R}$, then $\displaystyle\int\limits_{a}^{b}x^{n}\,dx=\int\limits_{0}^{b}x^{n}\,dx-\int% \limits_{0}^{a}x^{n}\,dx=\frac{1}{n+1}b^{{n+1}}-\frac{1}{n+1}a^{{n+1}}.$

It follows that $\displaystyle\int x^{n}\,dx=\frac{1}{n+1}x^{n}+C$. ∎

Type of Math Object:

Theorem

Major Section:

Reference

## Mathematics Subject Classification

26A42*no label found*

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## Comments

## TeX error

I am uncertain what is wrong with the TeX code I have put in. I will try to correct it as soon as possible. For now, hopefully you can tell how the proof is supposed to go from the TeX code.

## Re: TeX error

One problem is that the brackets don't match in the following fragment

\frac{B_{n+1-l}{n+1}(t+1)^l

There are a lot of = signs sitting outside maths environments as well, these ought to go inside the $...$s. And it would be better style to use a theorem/proof environment rather than \mathbf{Theorem: }

Also the comma ending the very last displayed equation: is that supposed to be there?

## Re: TeX error

Thanks for the \frac correction. Those types are the hardest for me to see. The = ended up outside of the $'s because I tried to use eqnarray and failed miserably, and I forgot to fix it afterwards. In any case, I'm glad that I finally got this debugged to the point that other people can read the html. Thanks.

## no more \mathbf{Theorem:}!

A long time ago, silverfish wrote:

> And it would be better style to use a theorem/proof

> environment rather than \mathbf{Theorem: }

Others have made similar comments.

Probably to the sheer delight of all of those nagging me about this, I am going back through and changing all of these.

The main reason that I was using \mathbf{Theorem:} was that I could not figure out how to suppress the numbering in the theorem/proof environment. To me, using the obscure \mathbf{Theorem:} was a far lesser evil than having numbered theorems in entries that contained only one theorem. Now that I have asterisks everywhere, life is good. :-)

## Re: no more \mathbf{Theorem:}!

I am sheerly delighted :)