integration of polynomial

Theorem.

For all nonnegative integers $n$,

$$\int x^n𝑑x=\frac{1}{n+1}{x}^{n+1}+C.$$

Proof.

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

$$\underset{0}{\overset{a}{\int }}{x}^n𝑑x=\frac{1}{n+1}{a}^{n+1}.$$

If $a=0$, the above statement is obvious. If $a>0$, the following computation uses the right hand rule for computing the integral (http://planetmath.org/RiemannIntegral); if , the following computation uses the left hand rule for computing the integral:

${\displaystyle \underset{0}{\overset{a}{\int }}}{x}^n𝑑x$	$=\underset{t\to \mathrm{\infty }}{lim}{\displaystyle \sum _{k=1}^t}{\left({\displaystyle \frac{ak}{t}}\right)}^n\left({\displaystyle \frac{a}{t}}\right)$
	$=a^{n+1}\underset{t\to \mathrm{\infty }}{lim}{\displaystyle \frac{1}{t^{n+1}}}{\displaystyle \sum _{k=1}^t}{k}^n$
	$=a^{n+1}\underset{t\to \mathrm{\infty }}{lim}{\displaystyle \frac{1}{t^{n+1}}}{\displaystyle \sum _{l=1}^{n+1}}\left({\displaystyle \genfrac{}{}{0pt}{}{n+1}{r}}\right){\displaystyle \frac{B_{n+1-l}}{n+1}}{(t+1)}^l$ by this theorem (http://planetmath.org/SumOfKthPowersOfTheFirstNPositiveIntegers),
	$=a^{n+1}\underset{t\to \mathrm{\infty }}{lim}{\displaystyle \frac{1}{t^{n+1}}}\left({\displaystyle \genfrac{}{}{0pt}{}{n+1}{n+1}}\right){\displaystyle \frac{B_{n+1-(n+1)}}{n+1}}{(t+1)}^{n+1}$
	$={\displaystyle \frac{B_0}{n+1}}{a}^{n+1}\underset{t\to \mathrm{\infty }}{lim}{\left({\displaystyle \frac{t+1}{t}}\right)}^{n+1}$
	$={\displaystyle \frac{1}{n+1}}{a}^{n+1}$

Thus, if $a,b\in ℝ$, then ${\displaystyle \underset{a}{\overset{b}{\int }}}{x}^n𝑑x={\displaystyle \underset{0}{\overset{b}{\int }}}{x}^n𝑑x-{\displaystyle \underset{0}{\overset{a}{\int }}}{x}^n𝑑x={\displaystyle \frac{1}{n+1}}{b}^{n+1}-{\displaystyle \frac{1}{n+1}}{a}^{n+1}.$

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

Title	integration of polynomial
Canonical name	IntegrationOfPolynomial
Date of creation	2013-03-22 15:57:29
Last modified on	2013-03-22 15:57:29
Owner	Wkbj79 (1863)
Last modified by	Wkbj79 (1863)
Numerical id	30
Author	Wkbj79 (1863)
Entry type	Theorem
Classification	msc 26A42