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Question Number 129048 by bramlexs22 last updated on 12/Jan/21

$$\mathrm{What}\mathrm{is}\mathrm{the}\mathrm{Laplace}\mathrm{transform}$$$$\mathrm{of}\mathrm{f}\left(\mathrm{t}\right)=-{4\mathrm{t}}^2-5\sin 3\mathrm{t}$$

Answered by Dwaipayan Shikari last updated on 12/Jan/21

$$\mathcal{L}\left(f\left(t\right)\right)=-4{\int }_0^{\mathrm{\infty }}{t}^2{e}^{-st}-5{\int }_0^{\mathrm{\infty }}{e}^{-st}sin3t$$$$=-\frac{4}{s^3}{\int }_0^{\mathrm{\infty }}{j}^2{e}^{-j}dj-\frac{5}{2i}{\int }_0^{\mathrm{\infty }}{e}^{-(s-3i)}-e^{-(s+3i)}dt$$$$=-\frac{8}{s^3}-\frac{5}{2i}(\frac{1}{s-3i}-\frac{1}{s+3i})=-\frac{8}{s^3}-\frac{15}{s^2+9}$$

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