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With the input

f[t_] := Cos[2 Pi t]
FourierTransform[f[t], t, w, FourierParameters -> {0, -2*Pi}]

I get 1/2 DiracDelta[-1 + w] + 1/2 DiracDelta[1 + w] as expected. But unless I'm having a brain fart, this implies that

Integrate[f[t]*Exp[-2 Pi I t 3], {t, -Infinity, Infinity}]

is zero. Mathematica, however, claims that this integral doesn't converge. I could understand if FourierTransform used other methods internally and thus Integrate doesn't find a solution. But I'm a bit surprised that I get the apodictic claim that the integral doesn't converge. What am I missing? Are there any options to Integrate or any assumptions I could use to make Mathematica change its mind?

BTW, I am aware of this, but although it's related it doesn't answer my question.

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closed as off-topic by Daniel Lichtblau, m_goldberg, mikado, Henrik Schumacher, LCarvalho Jan 3 at 19:07

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    $\begingroup$ What's strange here? The integral diverge? period. Fourier transform command deals with distributions and action of the FT on those are not integrating generally. It's a more widely defined mapping. $\endgroup$ – Andrew Dec 28 '18 at 17:49
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    $\begingroup$ Stated differently, not every FT can be recast as a convergent integral. $\endgroup$ – Daniel Lichtblau Dec 28 '18 at 21:23
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The reason is that Integrate will not produce generalized functions like DiracDelta as results. I don't believe there is a strategy that will produce consistent results in general for such (formally divergent) integrals. FourierTransform is restricted to Fourier integrals, a relatively safe domain.

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