integration of polynomial
Theorem.
For all nonnegative integers n,
∫xn𝑑x=1n+1xn+1+C. |
Proof.
It will first be proven that, for any nonnegative integer n and any a∈ℝ,
a∫0xn𝑑x=1n+1an+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 a<0, the following computation uses the left hand rule for computing the integral:
a∫0xn𝑑x | =lim |
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by this theorem (http://planetmath.org/SumOfKthPowersOfTheFirstNPositiveIntegers), | |
Thus, if , then
It follows that . ∎
Title | integration of polynomial |
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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 |