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Questions about the function Assuming[], the $Assumptions system variable, and the Assumptions option of functions such as Simplify[] and Integrate[].

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Identical integral evaluating differently

I am trying to compute an integral but am getting two different answers using two near-identical methods. I wish to know why they differ. f[x_] := x^2 + 1 g[x_] := 1/(2 π) 1/(x^2 + (1/4)) Simplify[ …