Question Number 64970 by mathmax by abdo last updated on 23/Jul/19 | ||
$${let}\:{f}\left({a}\right)=\int_{\mathrm{0}} ^{\infty} \:\:\frac{{cos}\left({x}^{\mathrm{2}} \right)\:+{sin}\left({x}^{\mathrm{2}} \right)}{\left({x}^{\mathrm{2}} \:+{a}^{\mathrm{2}} \right)^{\mathrm{2}} }\:{dx}\:\:\:{with}\:{a}>\mathrm{0} \\ $$ $$\left.\mathrm{1}\right)\:{calculate}\:{f}\left({a}\right) \\ $$ $$\left.\mathrm{2}\right)\:{find}\:{the}\:{values}\:{of}\:\int_{\mathrm{0}} ^{\infty} \:\:\frac{{cos}\left({x}^{\mathrm{2}} \right)+{sin}\left({x}^{\mathrm{2}} \right)}{\left({x}^{\mathrm{2}} +\mathrm{1}\right)^{\mathrm{2}} }{dx}\:{and} \\ $$ $$\int_{\mathrm{0}} ^{\infty} \:\frac{{cos}\left({x}^{\mathrm{2}} \right)+{sin}\left({x}^{\mathrm{2}} \right)}{\left({x}^{\mathrm{2}} +\mathrm{3}\right)^{\mathrm{2}} }{dx} \\ $$ | ||
Commented by~ À ® @ 237 ~ last updated on 23/Jul/19 | ||
$$ \\ $$ $$\:\:\:\:{we}\:\:{always}\:{have}\:\:{cos}\left({x}^{\mathrm{2}} \right)+{sin}\left({x}^{\mathrm{2}} \right)=\mathrm{1} \\ $$ $${so}\:\:{f}\left({a}\right)=\int_{\mathrm{0}} ^{\infty} \frac{\mathrm{1}}{\left({x}^{\mathrm{2}} +{a}^{\mathrm{2}} \right)^{\mathrm{2}} }{dx} \\ $$ $${let}\:{change}\:\:{x}\:={a}.{tant}\:\:\:\:\:\:\:\:{dx}\:={a}\:\left(\mathrm{1}+{tan}^{\mathrm{2}} {t}\right){dt} \\ $$ $$\:\:\:{f}\left({a}\right)\:=\:\int_{\mathrm{0}} ^{\frac{\pi}{\mathrm{2}}} \frac{{a}\left(\mathrm{1}+{tan}^{\mathrm{2}} {t}\right){dt}}{\left({a}^{\mathrm{2}} {tan}^{\mathrm{2}} {t}\:+{a}^{\mathrm{2}} \right)^{\mathrm{2}} } \\ $$ $$\:\:\:\:\:\:\:\:\:\:\:\:=\frac{\mathrm{1}}{{a}^{\mathrm{3}} }\:\int_{\mathrm{0}} ^{\frac{\pi}{\mathrm{2}}} \frac{\mathrm{1}}{\left(\mathrm{1}+{tan}^{\mathrm{2}} {t}\right)}{dt} \\ $$ $$\:\:\:\:\:\:\:=\frac{\mathrm{1}}{{a}^{\mathrm{3}} }\:\int_{\mathrm{0}\:\:\:} ^{\frac{\pi}{\mathrm{2}}} {cos}^{\mathrm{2}} {t}\:\:{dt}\: \\ $$ $${knowing}\:{that}\:\:{cos}^{\mathrm{2}} {t}\:\:=\:\:\frac{\mathrm{1}+{cos}\mathrm{2}{t}}{\mathrm{2}}\:\:\:{we}\:\:{finally}\:\:{got} \\ $$ $$\:\:\:\:{f}\left({a}\right)=\:\frac{\mathrm{1}}{{a}^{\mathrm{3}} }\:\left[\frac{{t}}{\mathrm{2}}\:+\frac{\mathrm{1}}{\mathrm{4}}{sin}\mathrm{2}{t}\right]_{\mathrm{0}} ^{\frac{\pi}{\mathrm{2}}} \\ $$ $$\:\:\:\:\:\:\:\:=\:\:\frac{\pi}{\mathrm{4}{a}^{\mathrm{3}} } \\ $$ $${then}\:\:\:{f}\left(\mathrm{1}\right)\:=\:\frac{\pi}{\mathrm{4}}\:\:\:\:\:\:{and}\:\:{f}\left(\sqrt{\mathrm{3}}\right)\:=\:\frac{\pi}{\mathrm{12}\sqrt{\mathrm{3}}} \\ $$ | ||
Commented bymathmax by abdo last updated on 23/Jul/19 | ||
$${thank}\:{you}\:{sir}. \\ $$ | ||
Commented bymathmax by abdo last updated on 23/Jul/19 | ||
$${really}\:{its}\:{cos}\left({x}^{\mathrm{2}} \right)−{sin}\left({x}^{\mathrm{2}} \right){not}\:+\:{but}\:{nevermind}\:{i}\:{will}\:{post} \\ $$ $${another}\:{question}... \\ $$ | ||
Commented byMJS last updated on 24/Jul/19 | ||
$$\left(\mathrm{cos}\:{x}\right)^{\mathrm{2}} +\left(\mathrm{sin}\:{x}\right)^{\mathrm{2}} =\mathrm{1} \\ $$ $$\mathrm{but} \\ $$ $$\mathrm{cos}\:\left({x}^{\mathrm{2}} \right)\:+\mathrm{sin}\:\left({x}^{\mathrm{2}} \right)\:=\sqrt{\mathrm{2}}\mathrm{sin}\:\left({x}^{\mathrm{2}} +\frac{\pi}{\mathrm{4}}\right) \\ $$ $$\mathrm{which}\:\mathrm{is}\:{not}\:\mathrm{always}\:=\mathrm{1} \\ $$ | ||
Commented bymathmax by abdo last updated on 24/Jul/19 | ||
$${sir}\:\sim\:\mathrm{237}\:\:\:{you}\:{answer}\:{is}\:{not}\:{correct}\:\:.... \\ $$ | ||
Commented bymathmax by abdo last updated on 24/Jul/19 | ||
$${you}\:{are}\:{right}\:{sir}\:\:{i}\:{have}\:{commited}\:{a}\:{error}\:{i}\:{delet}\:{this}\:{post}\: \\ $$ $${and}\:{give}\:{the}\:{right}\:{answer}\:... \\ $$ | ||
Commented bymathmax by abdo last updated on 24/Jul/19 | ||
$$\left.\mathrm{1}\right){we}\:{have}\:{cos}\left({x}^{\mathrm{2}} \right)+{sin}\left({x}^{\mathrm{2}} \right)\:=\sqrt{\mathrm{2}}{cos}\left({x}^{\mathrm{2}} −\frac{\pi}{\mathrm{4}}\right)\:\Rightarrow \\ $$ $${f}\left({a}\right)\:=\sqrt{\mathrm{2}}\int_{\mathrm{0}} ^{\infty} \:\:\:\frac{{cos}\left({x}^{\mathrm{2}} −\frac{\pi}{\mathrm{4}}\right)}{\left({x}^{\mathrm{2}} \:+{a}^{\mathrm{2}} \right)^{\mathrm{2}} }{dx}\:\Rightarrow\mathrm{2}{f}\left({a}\right)\:=\sqrt{\mathrm{2}}\int_{−\infty} ^{+\infty} \:\frac{{cos}\left({x}^{\mathrm{2}} −\frac{\pi}{\mathrm{4}}\right)}{\left({x}^{\mathrm{2}} \:+{a}^{\mathrm{2}} \right)^{\mathrm{2}} }{dx} \\ $$ $$\Rightarrow\sqrt{\mathrm{2}}{f}\left({a}\right)\:={Re}\left(\int_{−\infty} ^{+\infty} \:\:\frac{{e}^{{i}\left({x}^{\mathrm{2}} −\frac{\pi}{\mathrm{4}}\right)} }{\left({x}^{\mathrm{2}} \:+{a}^{\mathrm{2}} \right)^{\mathrm{2}} }{ex}\right)\:{let}\:\varphi\left({z}\right)\:=\frac{{e}^{{i}\left({z}^{\mathrm{2}} −\frac{\pi}{\mathrm{4}}\right)} }{\left({z}^{\mathrm{2}} \:+{a}^{\mathrm{2}} \right)^{\mathrm{2}} }\:\Rightarrow \\ $$ $$\varphi\left({z}\right)\:=\frac{{e}^{{i}\left({z}^{\mathrm{2}} −\frac{\pi}{\mathrm{4}}\right)} }{\left({z}−{ia}\right)^{\mathrm{2}} \left({z}+{ia}\right)^{\mathrm{2}} }\:\:{the}\:{poles}\:{of}\:\varphi\:{are}\:\overset{−} {+}{ia}\:\:\left({a}>\mathrm{0}\right)\:{residus} \\ $$ $${theorem}\:{give}\:\int_{−\infty} ^{+\infty} \:\varphi\left({z}\right){dz}\:=\mathrm{2}{i}\pi\:{Res}\left(\varphi,{ia}\right) \\ $$ $${Res}\left(\varphi,{ia}\right)\:={lim}_{{z}\rightarrow{ia}} \left({z}−{ia}\right)^{\mathrm{2}} \left\{\left({z}−{ia}\right)^{\mathrm{2}} \varphi\left({z}\right)\right\}^{\left(\mathrm{1}\right)} \\ $$ $$={lim}_{{z}\rightarrow{ia}} \:\:\left\{\frac{{e}^{{i}\left({z}^{\mathrm{2}} −\frac{\pi}{\mathrm{4}}\right)} }{\left({z}+{ia}\right)^{\mathrm{2}} }\right\}^{\left(\mathrm{1}\right)} \:={e}^{−\frac{{i}\pi}{\mathrm{4}}} \:{lim}_{{z}\rightarrow{ia}} \:\:\:\left\{\frac{{e}^{{iz}^{\mathrm{2}} } }{\left({z}+{ia}\right)^{\mathrm{2}} }\right\}^{\left(\mathrm{1}\right)} \\ $$ $$={e}^{−\frac{{i}\pi}{\mathrm{4}}} \:\:\:{lim}_{{z}\rightarrow{ia}} \:\:\:\frac{\mathrm{2}{iz}\:{e}^{{iz}^{\mathrm{2}} } \left({z}+{ia}\right)^{\mathrm{2}} \:−\mathrm{2}\left({z}+{ia}\right){e}^{{iz}^{\mathrm{2}} } }{\left({z}+{ia}\right)^{\mathrm{4}} } \\ $$ $$={e}^{−\frac{{i}\pi}{\mathrm{4}}} \:{lim}_{{z}\rightarrow{ia}} \:\:\:\:\frac{\left(\mathrm{2}{iz}\left({z}+{ia}\right)−\mathrm{2}\right){e}^{{iz}^{\mathrm{2}} } }{\left({z}+{ia}\right)^{\mathrm{3}} } \\ $$ $$={e}^{−{i}\frac{\pi}{\mathrm{4}}} \:\:\frac{\mathrm{2}{i}\left({ia}\right)\left(\mathrm{2}{ia}\right)−\mathrm{2}}{\left(\mathrm{2}{ia}\right)^{\mathrm{3}} }\:{e}^{{i}\left({ia}\right)^{\mathrm{2}} } =\:{e}^{−\frac{{i}\pi}{\mathrm{4}}} \:\:\frac{−\mathrm{4}{ia}^{\mathrm{2}} −\mathrm{2}}{−\mathrm{8}{ia}^{\mathrm{3}} }\:{e}^{−{ia}^{\mathrm{2}} } \\ $$ $$=\frac{\mathrm{2}{ia}^{\mathrm{2}} −\mathrm{1}}{\mathrm{4}{ia}^{\mathrm{3}} }\:{e}^{−{i}\left(\frac{\pi}{\mathrm{4}}+{a}^{\mathrm{2}} \right)} \:\Rightarrow\int_{−\infty} ^{+\infty} \:\varphi\left({z}\right){dz}\:=\mathrm{2}{i}\pi\:\frac{\mathrm{2}{ia}^{\mathrm{2}} −\mathrm{1}}{\mathrm{4}{ia}^{\mathrm{3}} }\:{e}^{−{i}\left(\frac{\pi}{\mathrm{4}}+{a}^{\mathrm{2}} \right)} \\ $$ $$=−\frac{\pi}{\mathrm{2}{a}^{\mathrm{3}} }\left(\mathrm{1}−\mathrm{2}{ia}^{\mathrm{2}} \right)\left(\:{cos}\left(\frac{\pi}{\mathrm{4}}+{a}^{\mathrm{2}} \right)−{isin}\left(\frac{\pi}{\mathrm{4}}+{a}^{\mathrm{2}} \right)\right) \\ $$ $$=−\frac{\pi}{\mathrm{2}{a}^{\mathrm{3}} }\left\{{cos}\left(\frac{\pi}{\mathrm{4}}+{a}^{\mathrm{2}} \right)−{isin}\left(\frac{\pi}{\mathrm{4}}+{a}^{\mathrm{2}} \right)−\mathrm{2}{ia}^{\mathrm{2}} {cos}\left(\frac{\pi}{\mathrm{4}}+{a}^{\mathrm{2}} \right)−\mathrm{2}{a}^{\mathrm{2}} {sin}\left(\frac{\pi}{\mathrm{4}}+{a}^{\mathrm{2}} \right)\right\} \\ $$ $$\sqrt{\mathrm{2}}{f}\left({a}\right)\:=−\frac{\pi}{\mathrm{2}{a}^{\mathrm{3}} }\left({cos}\left(\frac{\pi}{\mathrm{4}}+{a}^{\mathrm{2}} \right)−\mathrm{2}{a}^{\mathrm{2}} {sin}\left(\frac{\pi}{\mathrm{4}}\:+{a}^{\mathrm{2}} \right)\right) \\ $$ $$=−\frac{\pi}{\mathrm{2}{a}^{\mathrm{3}} }{cos}\left(\frac{\pi}{\mathrm{4}}+{a}^{\mathrm{2}} \right)\:+\frac{\pi}{{a}}\:{sin}\left(\frac{\pi}{\mathrm{4}}\:+{a}^{\mathrm{2}} \right)\:\Rightarrow \\ $$ $${f}\left({a}\right)\:=\frac{\pi}{{a}\sqrt{\mathrm{2}}}{sin}\left(\frac{\pi}{\mathrm{4}}\:+{a}^{\mathrm{2}} \right)−\frac{\pi}{\mathrm{2}\sqrt{\mathrm{2}}{a}^{\mathrm{3}} }\:{cos}\left(\frac{\pi}{\mathrm{4}}\:+{a}^{\mathrm{2}} \right) \\ $$ | ||
Commented bymathmax by abdo last updated on 24/Jul/19 | ||
$$\left.\mathrm{2}\right)\:\int_{\mathrm{0}} ^{\infty} \:\:\:\frac{{cos}\left({x}^{\mathrm{2}} \right)+{sin}\left({x}^{\mathrm{2}} \right)}{\left({x}^{\mathrm{2}} +\mathrm{1}\right)^{\mathrm{2}} }{dx}\:={f}\left(\mathrm{1}\right)\:=\frac{\pi}{\sqrt{\mathrm{2}}}{sin}\left(\frac{\pi}{\mathrm{4}}+\mathrm{1}\right)−\frac{\pi}{\mathrm{2}\sqrt{\mathrm{2}}}{cos}\left(\frac{\pi}{\mathrm{4}}+\mathrm{1}\right) \\ $$ $$\int_{\mathrm{0}} ^{\infty} \:\:\frac{{cos}\left({x}^{\mathrm{2}} \right)+{sin}\left({x}^{\mathrm{2}} \right)}{\left({x}^{\mathrm{2}} \:+\mathrm{3}\right)^{\mathrm{2}} }\:={f}\left(\sqrt{\mathrm{3}}\right)\:=\frac{\pi}{\sqrt{\mathrm{6}}}{sin}\left(\frac{\pi}{\mathrm{4}}+\mathrm{3}\right)−\frac{\pi}{\mathrm{2}\sqrt{\mathrm{2}}\mathrm{3}\sqrt{\mathrm{3}}}\:{cos}\left(\frac{\pi}{\mathrm{4}}+\mathrm{3}\right) \\ $$ $$=\frac{\pi}{\sqrt{\mathrm{6}}}{sin}\left(\mathrm{3}+\frac{\pi}{\mathrm{4}}\right)−\frac{\pi}{\mathrm{6}\sqrt{\mathrm{6}}}\:{cos}\left(\mathrm{3}+\frac{\pi}{\mathrm{4}}\right). \\ $$ | ||
Commented by~ À ® @ 237 ~ last updated on 25/Jul/19 | ||
$$\:\:\:{yes}\:\:{you}\:{are}\:{right}\:.\:{I}\:{did}\:{not}\:{mind}\:{the}\:{difference}\:{at}\:{this}\:{time}\:\:.\:{Sorry} \\ $$ | ||