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Why is this integral not convergent? Could be a technicality
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[QUOTE="Nathanael, post: 4989415, member: 509990"] The function is indeed exactly odd! (Check by the definition of odd, [itex]-y(-x)=y(x)[/itex]) But this is not enough when it comes to infinity. The other answers were interesting, and I don't have much insight to contribute, but I wanted to share a few sentences from my Calculus book (by Gilbert Strang). It says, "the (...) integral (...) from [itex]a=-∞[/itex] to [itex]b=∞[/itex] (...) is split into two parts, and [B][I]each part much converge. [/I][/B]By definition, the limits at -∞ and ∞ are kept [I]separate[/I]." (The bold and italics are from him not me.) He goes on to emphasize this point a little later with an odd function like yours: "[itex]\int_{-∞}^{∞}x.dx[/itex] [I]is not defined even though[/I] [itex]\int_{-b}^{b}x.dx=0[/itex] [I]for every b[/I]. The area under y=x is +∞ on one side of zero. The area is -∞ on the other side. [I][B]We cannot accept [/B][/I][itex]∞-∞=0[/itex]. The two areas must be separately finite, and in this case they are not." [/QUOTE]
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Why is this integral not convergent? Could be a technicality
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