fourier transformation of f(t)

Question

I stucking on this exercise.

Should I use any Fourier Transformation rules?

Thank you very much in advance for your answer ^^

Answer 1

\begin{aligned} f(t)&=(3e^{-2t^2}-5)\mathbf{1}_{(0,4)}(t) + 3e^{-2t^2}\mathbf{1}_{(0,4)^c}(t)\\ &= 3e^{-2t^2} - 5\mathbf{1}_{(0,5)}(t) \end{aligned}

Then, $\big(\mathcal{F}f\big)(s)=3\mathcal{F}(e^{-2t^2})(s)- 5\int^4_0e^{-2\pi st}\,dt$ The term on the right-hand side should be easily derived from the Fourier transform of a Gaussian function; the second is a direct computation.