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:

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