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

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

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Start with the definite integral.

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Inverse tangent form: the integral of du/(a squared plus u squared) equals (1/a) times arctan(u/a).

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Antiderivative.

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Apply the fundamental theorem of calculus.

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So the answer is 0.7853981634 equals pi/4.

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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 0.7853981634, which checks out.