If it is not a mathematical analysis problem, but a high number problem, you can set the known f derivative, and the derivative and integral can be interchanged. 1.f(ω)= 2∫_ { 0→∞} e[-(t/τ)2]cos(ωt)dt = = > f'(ω)=-2∫_{0→∞}e^[-(t/τ)^2]tsin(ωt)dt= =τ^2∫_{0→∞}sin(ωt)d{e^[-(t/τ)^2]}= =-τ^2ω∫_{0→∞}e^[-(t/τ)^2]cos(ωt)dt==-[τ^2ω/2]f(ω)2。 Solve the above differential equation: f (ω) = ce [-(τ 2ω 2/4)] c = f (0) = 2 ∫ _ {0 →∞} e [-(t/τ) 2] dt = = f (ω) = [τ√π] e [-. You can also use methods such as complex variables.