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Here is the problem: integrate x/(x squared plus 1) from 0 to 2.

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Evaluate the definite integral of x / (x squared plus 1) from x equals 0 to x equals 2.

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Start with the definite integral.

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Logarithm rule: the numerator is a constant multiple of the denominator prime s derivative, the integral of u prime /u dx equals ln|u|.

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Antiderivative.

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Apply the fundamental theorem of calculus.

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So the answer is 0.8047189562.

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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 0.8047189562, which checks out.