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Here is the problem: integral of x sin(x) from 0 to pi.

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Evaluate the definite integral of x sin(x) from x equals 0 to x equals pi.

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Start with the definite integral.

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Integration by parts (tabular method). Differentiate the polynomial x until it reaches 0, integrate sin(x) the same number of times, and combine with alternating signs.

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Combine the alternating products.

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Antiderivative.

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Apply the fundamental theorem of calculus.

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So the answer is 3.141592654 equals pi.

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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 3.141592654, which checks out.