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Here is the problem: integral of x cubed minus 2x plus 1.

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Find the indefinite integral (antiderivative) of x cubed minus 2 x plus 1 with respect to x.

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Start with the integral.

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Power rule: the integral of x to the power n dx equals x to the power n plus 1/(n plus 1).

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Power rule: the integral of x to the power n dx equals x to the power n plus 1/(n plus 1).

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Constant multiple rule: the integral of k times f dx equals k the integral of f dx.

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Sum rule: integrate term by term.

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Constant rule: the integral of k dx equals kx.

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Sum rule: integrate term by term.

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Simplify and add the constant of integration.

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So the answer is x to the power 4 over 4 plus x minus x squared plus C.

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An antiderivative F satisfies F prime of x equals f(x). Because the derivative of any constant is 0, every antiderivative differs by a constant C.

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Differentiated the answer and compared it with the integrand at 8 sample points, which checks out.