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Category: Integration

L-sinx-0-e-sx-sinx-dx-0-e-sx-e-ix-e-ix-2i-dx-1-2i-0-e-s-i-x-dx-0-e-s-i-x-dx-1-2i-1-s-i-e-s-i-x-1-s-i-e-s-i-x-0-1-

Question Number 219449 by Lukos last updated on 25/Apr/25 $${L}\left\{{sinx}\right\}=\int_{\mathrm{0}} ^{\infty} {e}^{−{sx}} {sinx}\:{dx}=\int_{\mathrm{0}} ^{\infty} {e}^{−{sx}} \frac{{e}^{{ix}} −{e}^{−{ix}} }{\mathrm{2}{i}}{dx} \\ $$$$=\frac{\mathrm{1}}{\mathrm{2}{i}}\left[\int_{\mathrm{0}} ^{\infty} {e}^{−\left({s}−{i}\right){x}} {dx}\:\:−\int_{\mathrm{0}} ^{\infty} {e}^{−\left({s}+{i}\right){x}}…