Integration Questions

Question Number 182891 by mathlove last updated on 16/Dec/22

$$\int_{\mathrm{0}\:\:} ^{\infty} {e}^{{x}} \frac{{sinmx}}{{x}}{dx}=? \\$$

Answered by dre23 last updated on 16/Dec/22

$${f}\left({t}\right)=\int_{\mathrm{0}} ^{\infty} {e}^{−{x}} .\frac{{sin}\left({tx}\right)}{{x}}{dx} \\$$ $${f}\left(\mathrm{0}\right)=\mathrm{0} \\$$ $${f}'\left({t}\right)=\int_{\mathrm{0}} ^{\infty} {e}^{−{x}} {cos}\left({xt}\right){dx} \\$$ $$={Re}\int_{\mathrm{0}} ^{\infty} {e}^{−\left(\mathrm{1}+{it}\right){x}} {dx},{x}>\mathrm{0} \\$$ $$\mathrm{1}+{it}=\sqrt{\mathrm{1}+{t}^{\mathrm{2}} }{e}^{{iarctan}\left({t}\right)} =\sqrt{\mathrm{1}+{t}^{\mathrm{2}} }{e}^{{ia}\left({t}\right)} \\$$ $$\left.{let}\:{C}=\left[\mathrm{0},{R}\right]\cup{Re}^{{i}\Omega} ,\Omega\in\left[−{a}\left({t},\right)\right]\cup{z}\:{e}^{−{ia}\left({t}\right)} ,{z}\in\left[\mathrm{0},{R}\right]\right\} \\$$ $$\int_{{C}} {e}^{−{x}} {cos}\left({xt}\right){dx}=\mathrm{0},\forall{t}>\mathrm{0}\:{holomorphic}\:{Function} \\$$ $$\int_{{Re}^{{i}\Omega\left({t}\right)} } {e}^{−{x}} {cos}\left({xt}\right){dx}=\mathrm{0} \\$$ $$\Leftrightarrow\int_{\mathrm{0}} ^{\infty} {e}^{−{x}} {cos}\left({xt}\right){dx}={Re}\left(\mathrm{1}−{it}\right)\int_{\mathrm{0}} ^{\infty} {e}^{−\left(\mathrm{1}+{it}\right)\left(\mathrm{1}−{it}\right){z}} {dz} \\$$ $$={Re}\left(\mathrm{1}−{it}\right)\int_{\mathrm{0}} ^{\infty} {e}^{−\left(\mathrm{1}+{t}^{\mathrm{2}} \right){z}} {dz}.=\frac{\mathrm{1}}{\mathrm{1}+{t}^{\mathrm{2}} } \\$$ $${f}'\left({t}\right)=\frac{\mathrm{1}}{\mathrm{1}+{t}^{\mathrm{2}} }\Rightarrow{f}\left({t}\right)={arctan}\left({t}\right)+{c} \\$$ $${f}\left(\mathrm{0}\right)=\mathrm{0}\Rightarrow{f}\left({t}\right)={arctan}\left({t}\left\{\right.\right. \\$$ $$\int_{\mathrm{0}} ^{\infty} {e}^{−{x}} .\frac{{sin}\left({mx}\right)}{{x}}{dx}={f}\left({m}\right)={arctan}\left({m}\right),\forall{m}\geqslant\mathrm{0} \\$$ $$\\$$

Commented byMl last updated on 16/Dec/22

$$\mathrm{why}\:\mathrm{you}\:\mathrm{e}^{\mathrm{x}} =\mathrm{e}^{−\mathrm{x}} \\$$

Commented byPeace last updated on 16/Dec/22

$${intial}\:\int{e}^{{x}} \frac{{sin}\left({mx}\right)}{{x}}{dx}\:{diverge} \\$$