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Prove that an antiderivative of a polynomial $f(x)=a_nx^n+a_{n-1}x^{n-1}+\cdots +a_0$ is $\begin{eqnarray} F(x)&=&\frac{a_n}{n+1}x^{n+1}+\frac{a_{n-1}}{n}x^n\\ &+&\cdots +a_0x+C. \end{eqnarray}$

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Lesson Parent:

Problem:

Prove that an antiderivative of a polynomial $f(x)=a_nx^n+a_{n-1}x^{n-1}+\cdots +a_0$ is $F(x)=\frac{a_n}{n+1}x^{n+1}+\frac{a_{n-1}}{n}x^n+\cdots +a_0x+C.$

Answer:

It is true that an antiderivative of a polynomial $f(x)=a_nx^n+a_{n-1}x^{n-1}+\cdots +a_0$ is $F(x)=\frac{a_n}{n+1}x^{n+1}+\frac{a_{n-1}}{n}x^n+\cdots +a_0x+C.$