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1.47 An integral from the theory of differential equations

To show that, when $s>0$ and $a\in{\mathbb{R}}$ ,

F(s)=\int_{0}^{\infty}\sin ax\,e^{-sx}\,dx={{a}\over{s^{2}+a^{2}}}.

This integral is important in the theory of differential equations. (See Laplace transforms in slides 1.59-62 and 6.38-51.)

Solution. We first calculate the indefinite integral:

{I=\int\sin ax\,e^{-sx}\,dx=}{-\frac{1}{s}\sin ax\,e^{-sx}+\frac{a}{s}\int\cos ax% \,e^{-sx}\,dx}

{=-\frac{1}{s}\sin ax\,e^{-sx}-\frac{a}{s^{2}}\cos ax\,e^{-sx}-\frac{a^{2}}{s^% {2}}\int\sin ax\,e^{-sx}\,dx}