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Here is the problem: integrate x cubed minus 2x from 0 to 4.

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Evaluate the definite integral of x cubed minus 2 x from x equals 0 to x equals 4.

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Start with the definite 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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Antiderivative.

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Apply the fundamental theorem of calculus.

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So the answer is 48.

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A definite integral gives the signed area between the curve and the x-axis: area above the axis counts as positive and area below counts as negative.

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Differentiated the antiderivative and compared it with the integrand, which checks out.

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Adaptive Simpson prime s rule gives 48, which checks out.