Question Number 27693 by abdo imad last updated on 12/Jan/18
$$\left.\mathrm{1}\right)\:{calculate}\:\:\int\int_{\left.\right]\left.\mathrm{0}\left.,\left.\mathrm{1}\right]×\right]\mathrm{0},\frac{\pi}{\mathrm{2}}\right]} \:\:\:\frac{{dxdy}}{\mathrm{1}+\left({xtany}\right)^{\mathrm{2}} } \\ $$$$\left.\mathrm{2}\right)\:{find}\:{the}\:{value}\:{of}\:\:\int_{\mathrm{0}} ^{\frac{\pi}{\mathrm{2}}} \:\:\frac{{t}}{{tant}}{dt}\:. \\ $$
Commented by abdo imad last updated on 13/Jan/18
$$\left.\mathrm{1}\right){let}\:{put}\:{I}=\:\int\int_{\left.\right]\left.\mathrm{0}\left.,\left.\mathrm{1}\right]×\right]\mathrm{0},\frac{\pi}{\mathrm{2}}\right]} \:\:\frac{{dxdy}}{\mathrm{1}+\left({xtany}\right)^{\mathrm{2}} } \\ $$$${I}=\:\int_{\mathrm{0}} ^{\mathrm{1}} \:\:\left(\int_{\mathrm{0}} ^{\frac{\pi}{\mathrm{2}}} \:\frac{{dy}}{\mathrm{1}+{x}^{\mathrm{2}} {tan}^{\mathrm{2}} {y}}\right){dx}\:\:\:{but}\:{the}\:{ch}.\:{tany}={t}\:{give} \\ $$$$\int_{\mathrm{0}} ^{\frac{\pi}{\mathrm{2}}} \:\:\frac{{dy}}{\mathrm{1}+{x}^{\mathrm{2}} {tan}^{\mathrm{2}} {y}}\:=\:\int_{\mathrm{0}} ^{\infty} \:\frac{\mathrm{1}}{\left(\mathrm{1}+{x}^{\mathrm{2}} {t}^{\mathrm{2}} \right)}\:\frac{{dt}}{\left(\mathrm{1}+{t}^{\mathrm{2}} \right)} \\ $$$$=\:\frac{\mathrm{1}}{\mathrm{2}}\:\int_{{R}} \:\:\:\:\frac{{dt}}{\left(\mathrm{1}+{t}^{\mathrm{2}} \right)\left(\mathrm{1}+{x}^{\mathrm{2}} {t}^{\mathrm{2}} \right)}\:{let}\:{introduce}\:{the}\:{complex}\:{function} \\ $$$${f}\left({z}\right)=\:\:\:\frac{\mathrm{1}}{\left(\mathrm{1}+{z}^{\mathrm{2}} \right)\left(\mathrm{1}+{x}^{\mathrm{2}} {z}^{\mathrm{2}} \right)} \\ $$$${f}\left({z}\right)=\:\:\:\frac{\mathrm{1}}{{x}^{\mathrm{2}} \left({z}−{i}\right)\left({z}+{i}\right)\left({z}−\frac{{i}}{{x}}\right)\left({z}+\frac{{i}}{{x}}\right)}\left(\:\:{dont}\:{forget}\:{that}\:\mathrm{0}<{x}\leqslant\mathrm{1}\right) \\ $$$${the}\:{poles}\:{of}\:{f}\:{are}\:{i}\:{and}\:−{i}\:{and}\:\frac{{i}}{{x}}\:{and}\:−\frac{{i}}{{x}}\:\:{so} \\ $$$$\int_{{R}} \:\:\:{f}\left({z}\right){dz}=\mathrm{2}{i}\pi\left(\:\:{Re}\left({f},{i}\right)\:+{Re}\left({f},\frac{{i}}{{x}}\right)\right)\:{but} \\ $$$${Res}\left({f},{i}\right)={lim}_{{z}−>{i}} \left({z}−{i}\right){f}\left({z}\right)=\:\:\frac{\mathrm{1}}{{x}^{\mathrm{2}} \left(\mathrm{2}{i}\right)\left(−\mathrm{1}+\frac{\mathrm{1}}{{x}^{\mathrm{2}} }\right)} \\ $$$$=\:\frac{\mathrm{1}}{\mathrm{2}{i}\left(\mathrm{1}−{x}^{\mathrm{2}} \right)} \\ $$$${Re}\left({f},\:\frac{{i}}{{x}}\right)={lim}_{{z}−>\frac{{i}}{{x}}} \:\left({z}−\frac{{i}}{{x}}\right){f}\left({z}\right)=\:\:\:\:\:\:\:\frac{\mathrm{1}}{{x}^{\mathrm{2}} \left(−\frac{\mathrm{1}}{{x}^{\mathrm{2}} }+\mathrm{1}\right)\left(\frac{\mathrm{2}{i}}{{x}}\right)} \\ $$$$=\:\:\frac{{x}}{\mathrm{2}{i}\left({x}^{\mathrm{2}} −\mathrm{1}\right)} \\ $$$$\int_{{R}} {f}\left({z}\right){dz}=\:\mathrm{2}{i}\pi\left(\:\:\:\frac{\mathrm{1}}{\mathrm{2}{i}\left(\mathrm{1}−{x}^{\mathrm{2}} \right)}\:−\:\frac{{x}}{\mathrm{2}{i}\left(\mathrm{1}−{x}^{\mathrm{2}} \right)}\right) \\ $$$$=\:\pi\frac{\mathrm{1}−{x}}{\mathrm{1}−{x}^{\mathrm{2}} }\:=\:\frac{\pi}{\mathrm{1}+{x}}\:\:{and}\:\int_{\mathrm{0}} ^{\frac{\pi}{\mathrm{2}}} \:\:\:\frac{{dy}}{\mathrm{1}+{x}^{\mathrm{2}} {tan}^{\mathrm{2}} {y}}\:=\:\frac{\pi}{\mathrm{2}\left(\mathrm{1}+{x}\right)} \\ $$$$\Rightarrow\:{I}=\:\int_{\mathrm{0}} ^{\mathrm{1}} \:\frac{\pi}{\mathrm{2}\left(\mathrm{1}+{x}\right)}\:{dx}\:=\frac{\pi}{\mathrm{2}}\:\left[{ln}/\mathrm{1}+{x}/\right]_{\mathrm{0}} ^{\mathrm{1}} =\:\frac{\pi}{\mathrm{2}}\:{ln}\mathrm{2} \\ $$$$\left.\mathrm{2}\right){by}\:{fubini}\:{therem}\:{we}\:{have}\:{also}\: \\ $$$${I}=\int_{\mathrm{0}} ^{\frac{\pi}{\mathrm{2}}} \left(\:\int_{\mathrm{0}} ^{\mathrm{1}} \:\:\frac{{dx}}{\mathrm{1}+\left({xtany}\right)^{\mathrm{2}} }\right){dy}\:{and}\:{the}\:{ch}.\:{xtany}={t}\:{give} \\ $$$$\int_{\mathrm{0}} ^{\mathrm{1}} \:\:\:\:\frac{{dx}}{\mathrm{1}+\left({xtany}\right)^{\mathrm{2}} }=\:\int_{\mathrm{0}} ^{{tany}} \:\:\:\frac{\mathrm{1}}{\mathrm{1}+{t}^{\mathrm{2}} }\:\frac{{dt}}{{tany}} \\ $$$$=\frac{\mathrm{1}}{{tany}}\:{arctan}\left({tany}\right)=\:\frac{{y}}{{tany}}\:{so} \\ $$$${I}=\:\int_{\mathrm{0}} ^{\frac{\pi}{\mathrm{2}}} \:\frac{{y}}{{tany}}{dy}\:\:\Rightarrow\int_{\mathrm{0}} ^{\frac{\pi}{\mathrm{2}}} \:\:\:\frac{{t}}{{tant}}\:{dt}\:=\:\frac{\pi}{\mathrm{2}}\:{ln}\left(\mathrm{2}\right). \\ $$
Answered by sma3l2996 last updated on 13/Jan/18
$$\mathrm{1}:\:\:{I}=\int\int_{\left.\right]\left.\mathrm{0}\left.,\left.\mathrm{1}\right]×\right]\mathrm{0},\pi/\mathrm{2}\right]} \frac{{dxdy}}{\left({xtany}\right)^{\mathrm{2}} }=\underset{{b}\rightarrow\mathrm{0}} {{lim}}\int_{{b}} ^{\pi/\mathrm{2}} \left(\underset{{a}\rightarrow\mathrm{0}} {{lim}}\int_{{a}} ^{\mathrm{1}} \frac{{dx}}{\mathrm{1}+\left({xtany}\right)^{\mathrm{2}} }\right){dy} \\ $$$${let}\:\:{t}={xtany}\Rightarrow{dt}={tanydx} \\ $$$${I}=\underset{{b}\rightarrow\mathrm{0}} {{lim}}\int_{{b}} ^{\pi/\mathrm{2}} \frac{\mathrm{1}}{{tany}}×\underset{{a}\rightarrow\mathrm{0}} {{lim}}\left(\int_{{a}} ^{{tany}} \frac{{dt}}{\mathrm{1}+{t}^{\mathrm{2}} }\right){dy} \\ $$$$=\underset{{b}\rightarrow\mathrm{0}} {{lim}}\int_{{b}} ^{\pi/\mathrm{2}} \frac{\mathrm{1}}{{tany}}×\underset{{a}\rightarrow\mathrm{0}} {{lim}}\left[{tan}^{−\mathrm{1}} \left({t}\right)\right]_{{a}} ^{{tany}} {dy} \\ $$$$=\underset{{b}\rightarrow\mathrm{0}} {{lim}}\int_{{b}} ^{\pi/\mathrm{2}} \frac{\mathrm{1}}{{tany}}\left({y}−\mathrm{0}\right){dy}=\underset{{b}\rightarrow\mathrm{0}} {{lim}}\int_{\mathrm{0}} ^{\pi/\mathrm{2}} \frac{{ydy}}{{tany}} \\ $$$${let}\:{z}={e}^{{iy}} \\ $$$${I}=−\int_{\Delta} \frac{{ln}\left({z}\right)\left({z}^{\mathrm{2}} −\mathrm{1}\right)}{{z}^{\mathrm{2}} +\mathrm{1}}×\frac{{dz}}{{z}} \\ $$$${let}\:{f}\left({z}\right)=\frac{{ln}\left({z}\right)\left({z}^{\mathrm{2}} −\mathrm{1}\right)}{{z}\left({z}+{i}\right)\left({z}−{i}\right)} \\ $$$${so}\:\:{I}=−\mathrm{2}{i}\pi{Res}\left({f},{i}\right)=−\mathrm{2}{i}\pi\left(\frac{{ln}\left({i}\right)\left(−\mathrm{2}\right)}{−\mathrm{2}}\right) \\ $$$${I}=\pi^{\mathrm{2}} \\ $$
Commented by abdo imad last updated on 13/Jan/18
$${the}\:{value}\:{of}\:\:{I}\:{you}\:{have}\:{given}\:{is}\:{not}\:{true}\:{sir}….{perhaps}\:{you}\:{have}\:{commiteted}\:{a}\:{error} \\ $$$${in}\:{residus}\:{application}…. \\ $$$$ \\ $$