What is the antiderivative of a polynomial?

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What is the antiderivative of a polynomial?

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Answer 1 · 2014-10-17T01:22:40

Let

$f (x) = a_{n} x^{n} + a_{n - 1} x^{n - 1} + \dots + a_{1} x + a_{0}$ $f(x)=a_nx^n+a_{n-1}x^{n-1}+cdots+a_1x+a_0$ .

An antiderivative $F (x)$ $F(x)$ of $f (x)$ $f(x)$ can be found by

$F (x) = \int f (x) d x$ $F(x)=int f(x)dx$

$= \int (a_{n} x^{n} + a_{n - 1} x^{n - 1} + \dots + a_{1} x + a_{0}) d x$ $=int(a_nx^n+a_{n-1}x^{n-1}+cdots+a_1x+a_0)dx$

$= \frac{a_{n}}{n + 1} x^{n + 1} + \frac{a_{n - 1}}{n} x^{n} + \dots + \frac{a_{1}}{2} x^{2} + a_{0} x + C$ $=a_n/{n+1}x^{n+1}+a_{n-1}/nx^n+cdots+a_1/2x^2+a_0x+C$ .

I hope that this was helpful.

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What is the antiderivative of a polynomial?

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