I would like to integrate the following product of Exp and Cos：
Integrate[Exp[-a x^2 + b x] Cos[x t], {x, -Infinity, Infinity}]
But the output is the same as the input form. Is it possible for Mathematica to perform this integration?
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1 Answer
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Besides of the method provide by @Michael E2,we can test the result by N-L formula.
int = Integrate[Exp[-a x^2 + b x] Cos[x t], x, Assumptions -> a > 0]
expr = Limit[int, x -> ∞, Assumptions -> a > 0] -
Limit[int, x -> -∞, Assumptions -> a > 0]
Simplify[expr // ComplexExpand, a > 0]
The same as
Assuming[a > 0,
Integrate[
Exp[-a x^2 + b x] Cos[x t] // TrigToExp, {x, -∞,∞}] // ComplexExpand // Simplify]
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$\begingroup$ Why this code does not work well? Integrate[Exp[-a x^2 + b x] Cos[x t], {x, -Infinity, Infinity}, Assumptions -> a > 0] // ComplexExpand // Simplify $\endgroup$ Commented May 16, 2022 at 11:03
Integrate[Exp[-a x^2 + b x] Cos[x t], {x, -Infinity, Infinity}, Assumptions -> t > 0 && a > 0 && b > 0]$\endgroup$res = Integrate[Exp[-a x^2 + b x] Cos[x t] // TrigToExp, {x, -Infinity, Infinity}, Assumptions -> a > 0]. Andres // ExpToTrig // TrigExpand // Simplifygives a nice looking result. $\endgroup$Integrate[ Exp[-a x^2 + b x] Cos[x t] // TrigToExp, {x, -Infinity, Infinity}]$\endgroup$