Question Number 197360 by hardmath last updated on 14/Sep/23
$$\mathrm{Find}: \\ $$$$\int_{\mathrm{0}} ^{\:\infty} \:\mathrm{sin}^{\mathrm{2}} \:\left(\:\sqrt{\mathrm{x}}\:\right)\:\mathrm{e}^{−\boldsymbol{\mathrm{x}}} \:\mathrm{dx}\:=\:? \\ $$
Answered by Mathspace last updated on 15/Sep/23
$${I}=\int_{\mathrm{0}} ^{\infty} \frac{\mathrm{1}−{cos}\left(\mathrm{2}\sqrt{{x}}\right)}{\mathrm{2}}\:{e}^{−{x}} {dx} \\ $$$$=\frac{\mathrm{1}}{\mathrm{2}}\int_{\mathrm{0}} ^{\infty} {e}^{−{x}} {dx}−\frac{\mathrm{1}}{\mathrm{2}}\int_{\mathrm{0}} ^{\infty} {cos}\left(\mathrm{2}\sqrt{{x}}\right){e}^{−{x}} {dx} \\ $$$$\int_{\mathrm{0}} ^{\infty} {e}^{−{x}} {dx}=\left[−{e}^{−{x}} \right]_{\mathrm{0}} ^{\infty} =\mathrm{1} \\ $$$$\int_{\mathrm{0}} ^{\infty} {cos}\left(\mathrm{2}\sqrt{{x}}\right){e}^{−{x}} {dx}\:\:\:\:\left(\sqrt{{x}}={t}\right) \\ $$$$=\int_{\mathrm{0}} ^{\infty} {cos}\left(\mathrm{2}{t}\right){e}^{−{t}^{\mathrm{2}} } \left(\mathrm{2}{t}\right){dt} \\ $$$$=\left[−{e}^{−{x}^{\mathrm{2}} } {cos}\left(\mathrm{2}{t}\right)\right]_{\mathrm{0}} ^{\infty} −\mathrm{2}\int_{\mathrm{0}} ^{\infty} {e}^{−{t}^{\mathrm{2}} } \:{sin}\left(\mathrm{2}{t}\right){dt} \\ $$$$=\mathrm{1}−\mathrm{2}\int_{\mathrm{0}} ^{\infty} \:{e}^{−{t}^{\mathrm{2}} } {sin}\left(\mathrm{2}{t}\right){dt} \\ $$$${but}\:\int_{\mathrm{0}} ^{\infty} {e}^{−{t}^{\mathrm{2}} } {sin}\left(\mathrm{2}{t}\right){dt} \\ $$$$={Im}\left(\int_{\mathrm{0}} ^{\infty} \:{e}^{−{t}^{\mathrm{2}} +\mathrm{2}{it}} {dt}\right) \\ $$$${and}\:\int_{\mathrm{0}} ^{\infty} {e}^{−{t}^{\mathrm{2}} +\mathrm{2}{it}} {dt}=\int_{\mathrm{0}} ^{\infty} {e}^{−\left({t}^{\mathrm{2}} −\mathrm{2}{it}\:+{i}^{\mathrm{2}} −{i}^{\mathrm{2}} \right)} \:{dt} \\ $$$$=\int_{\mathrm{0}} ^{\infty} \:{e}^{−\left({t}−{i}\right)^{\mathrm{2}} +\mathrm{1}} {dt}\:\:\:\left({t}−{i}={y}\right) \\ $$$$={e}\:\int_{−{i}} ^{\infty} \:{e}^{−{y}^{\mathrm{2}} } {dy}={e}\lambda_{\mathrm{0}} {erf}\left(−{i}\right)\Rightarrow \\ $$$$\int_{\mathrm{0}} ^{\infty} {sin}^{\mathrm{2}} \left(\sqrt{{x}}\right){e}^{−{x}} {dx} \\ $$$$=\frac{\mathrm{1}}{\mathrm{2}}−\frac{\mathrm{1}}{\mathrm{2}}\left\{\mathrm{1}−\mathrm{2}\:{Im}\left({e}\lambda_{{o}} {erf}\left(−{i}\right)\right\}\right. \\ $$$$={Im}\left({e}\lambda_{\mathrm{0}} {erf}\left(−{i}\right)\right\} \\ $$$$….{be}\:{continued}… \\ $$