Question Number 33259 by prof Abdo imad last updated on 14/Apr/18
$${find}\:{the}\:{value}\:{of}\:\:\int_{\mathrm{0}} ^{\infty} \:\:\:\frac{{arctan}\left(\mathrm{2}{x}\right)}{{a}^{\mathrm{2}} \:+{x}^{\mathrm{2}} }\:{dx}\:{with}\:{a}\neq\mathrm{0} \\ $$
Commented by prof Abdo imad last updated on 27/Apr/18
$${let}\:{put}\:\:{I}=\:\int_{\mathrm{0}} ^{\infty} \:\:\frac{{arctan}\left(\mathrm{2}{x}\right)}{{a}^{\mathrm{2}} \:+{x}^{\mathrm{2}} }{dx} \\ $$$${if}\:{a}>\mathrm{0}\:\:{ch}.{x}={at}\:{give}\: \\ $$$${I}\:=\:\int_{\mathrm{0}} ^{\infty} \:\:\frac{{arctan}\left(\mathrm{2}{at}\right)}{{a}^{\mathrm{2}} \left(\mathrm{1}+{t}^{\mathrm{2}} \right)}\:{adt}\:=\:\frac{\mathrm{1}}{{a}}\int_{\mathrm{0}} ^{\infty} \:\:\:\frac{{arctan}\left(\mathrm{2}{at}\right)}{\mathrm{1}+{t}^{\mathrm{2}} }{dt} \\ $$$$\Rightarrow{aI}\:=\:\int_{\mathrm{0}} ^{\infty} \:\:\frac{{arctan}\left(\mathrm{2}{at}\right)}{\mathrm{1}+{t}^{\mathrm{2}} }{dt}\:={f}\left({a}\right)\:{we}\:{have} \\ $$$${f}^{'} \left({a}\right)\:=\:\int_{\mathrm{0}} ^{\infty} \:\:\:\:\:\:\:\:\frac{\mathrm{2}{t}}{\left(\mathrm{1}+\mathrm{4}{a}^{\mathrm{2}} {t}^{\mathrm{2}} \right)\left(\mathrm{1}+{t}^{\mathrm{2}} \right)}{dt}\:\:{let}\:{decompose} \\ $$$${F}\left({t}\right)\:=\:\:\frac{\mathrm{2}{t}}{\left(\mathrm{1}+\mathrm{4}{a}^{\mathrm{2}} {t}^{\mathrm{2}} \right)\left(\mathrm{1}+{t}^{\mathrm{2}} \right)}\:=\:\frac{\alpha{t}\:+{b}}{{t}^{\mathrm{2}} +\mathrm{1}}\:+\:\frac{{ct}\:+{d}}{\mathrm{4}{a}^{\mathrm{2}} {t}^{\mathrm{2}} \:+\mathrm{1}} \\ $$$${F}\left(−{t}\right)=−{F}\left({t}\right)\:\Rightarrow\frac{−\alpha{t}+{b}}{{t}^{\mathrm{2}} +\mathrm{1}}\:+\frac{−{ct}\:+{d}}{\mathrm{4}{a}^{\mathrm{2}} {t}^{\mathrm{2}} +\mathrm{1}} \\ $$$$=\frac{−\alpha{t}−{b}}{{t}^{\mathrm{2}} \:+\mathrm{1}}\:+\frac{−{ct}\:−{d}}{\mathrm{4}{a}^{\mathrm{2}} {t}^{\mathrm{2}} \:+\mathrm{1}}\:\Rightarrow\:{b}={d}=\mathrm{0}\:\Rightarrow \\ $$$${F}\left({t}\right)=\:\frac{\alpha{t}}{{t}^{\mathrm{2}} +\mathrm{1}}\:\:+\frac{{ct}}{\mathrm{4}{a}^{\mathrm{2}} {t}^{\mathrm{2}} \:+\mathrm{1}} \\ $$$${lim}_{{t}\rightarrow+\infty} {t}\:{F}\left({t}\right)\:=\mathrm{0}\:=\alpha\:+\frac{{c}}{\mathrm{4}{a}^{\mathrm{2}} }\:\Rightarrow\mathrm{4}{a}^{\mathrm{2}} \alpha\:+{c}\:=\mathrm{0}\:\Rightarrow \\ $$$${c}=−\mathrm{4}{a}^{\mathrm{2}} \alpha\:\Rightarrow{F}\left({t}\right)=\:\frac{\alpha{t}}{{t}^{\mathrm{2}} +\mathrm{1}}\:−\mathrm{4}{a}^{\mathrm{2}} \:\:\frac{\alpha{t}}{\mathrm{4}{a}^{\mathrm{2}} {t}^{\mathrm{2}} +\mathrm{1}} \\ $$$${F}\left(\mathrm{1}\right)\:=\:\frac{\mathrm{2}}{\left(\mathrm{1}+\mathrm{4}{a}^{\mathrm{2}} \right)\mathrm{2}}\:=\frac{\mathrm{1}}{\mathrm{4}{a}^{\mathrm{2}} +\mathrm{1}}\:=\:\frac{\alpha}{\mathrm{2}}\:−\frac{\mathrm{4}{a}^{\mathrm{2}} \alpha}{\mathrm{4}{a}^{\mathrm{2}} +\mathrm{1}}\:\Rightarrow \\ $$$$\mathrm{1}\:=\frac{\mathrm{1}}{\mathrm{2}}\left(\mathrm{4}{a}^{\mathrm{2}} +\mathrm{1}\right)\alpha\:−\mathrm{4}{a}^{\mathrm{2}} \alpha\:=\left(\mathrm{2}{a}^{\mathrm{2}} \:+\frac{\mathrm{1}}{\mathrm{2}}−\mathrm{4}{a}^{\mathrm{2}} \right)\alpha \\ $$$$=\left(\frac{\mathrm{1}}{\mathrm{2}}\:−\mathrm{2}{a}^{\mathrm{2}} \right)\alpha\:=\frac{\mathrm{1}−\mathrm{4}{a}^{\mathrm{2}} }{\mathrm{2}}\:\alpha\:\Rightarrow\alpha=\:\frac{\mathrm{2}}{\mathrm{1}−\mathrm{4}{a}^{\mathrm{2}} } \\ $$$${F}\left({t}\right)\:=\frac{\mathrm{2}}{\mathrm{1}−\mathrm{4}{a}^{\mathrm{2}} }\:\frac{{t}}{{t}^{\mathrm{2}} \:+\mathrm{1}}\:−\mathrm{4}{a}^{\mathrm{2}} \:\frac{\mathrm{2}}{\mathrm{1}−\mathrm{4}{a}^{\mathrm{2}} }\:\:\frac{{t}}{\mathrm{4}{a}^{\mathrm{2}} {t}^{\mathrm{2}} \:+\mathrm{1}} \\ $$$${F}\left({t}\right)\:=\:\frac{\mathrm{2}}{\mathrm{1}−\mathrm{4}{a}^{\mathrm{2}} }\:\frac{{t}}{{t}^{\mathrm{2}} +\mathrm{1}}\:−\frac{\mathrm{8}{a}^{\mathrm{2}} }{\mathrm{1}−\mathrm{4}{a}^{\mathrm{2}} }\:\:\frac{{t}}{\mathrm{4}{a}^{\mathrm{2}} {t}^{\mathrm{2}} +\mathrm{1}} \\ $$$${f}^{'} \left({a}\right)\:=\:\frac{\mathrm{1}}{\mathrm{1}−\mathrm{4}{a}^{\mathrm{2}} }\:\int_{\mathrm{0}} ^{\infty} \:\frac{\mathrm{2}{tdt}}{{t}^{\mathrm{2}} \:+\mathrm{1}}\:\:−\frac{\mathrm{8}{a}^{\mathrm{2}} }{\mathrm{1}−\mathrm{4}{a}^{\mathrm{2}} }\int_{\mathrm{0}} ^{\infty} \:\frac{{tdt}}{\mathrm{4}{a}^{\mathrm{2}} {t}^{\mathrm{2}} \:+\mathrm{1}} \\ $$$${but}\:\:\int_{\mathrm{0}} ^{\infty} \:\:\frac{{tdt}}{\mathrm{4}{a}^{\mathrm{2}} {t}^{\mathrm{2}} \:+\mathrm{1}}\:=\frac{\mathrm{1}}{\mathrm{8}{a}^{\mathrm{2}} }\int_{\mathrm{0}} ^{\infty} \:\frac{\mathrm{8}{a}^{\mathrm{2}} {t}}{\mathrm{4}{a}^{\mathrm{2}} {t}^{\mathrm{2}} \:+\mathrm{1}} \\ $$$${f}^{'} \left({a}\right)=\:\frac{\mathrm{1}}{\mathrm{1}−\mathrm{4}{a}^{\mathrm{2}} }\left[{ln}\left(\frac{\mathrm{1}+{t}^{\mathrm{2}} }{\mathrm{4}{a}^{\mathrm{2}} {t}^{\mathrm{2}} +\mathrm{1}}\right)\right]_{\mathrm{0}} ^{+\infty} \:=\frac{\mathrm{1}}{\mathrm{1}−\mathrm{4}{a}^{\mathrm{2}} }{ln}\left(\frac{\mathrm{1}}{\mathrm{4}{a}^{\mathrm{2}} }\right) \\ $$$$=\:\frac{−{ln}\left(\mathrm{4}{a}^{\mathrm{2}} \right)}{\mathrm{1}−\mathrm{4}{a}^{\mathrm{2}} }\:\Rightarrow{f}\left({a}\right)\:=\:\int_{\mathrm{0}} ^{{a}} \:\:\frac{−{ln}\left(\mathrm{4}{x}^{\mathrm{2}} \right)}{\mathrm{1}−\mathrm{4}{x}^{\mathrm{2}} }{dx}\:+\lambda \\ $$$${but}\:\lambda\:={f}\left(\mathrm{0}\right)=\mathrm{0}\:\Rightarrow{f}\left({a}\right)\:=−\int_{\mathrm{0}} ^{{a}\:} \frac{\mathrm{2}{ln}\left(\mathrm{2}{x}\right)}{\mathrm{1}−\mathrm{4}{x}^{\mathrm{2}} }{dx} \\ $$$$−{f}\left({a}\right)=_{\mathrm{2}{x}={t}} \:\:\mathrm{2}\:\int_{\mathrm{0}} ^{\mathrm{2}{a}} \:\:\:\frac{{ln}\left({t}\right)}{\mathrm{1}−{t}^{\mathrm{2}} }\:\frac{{dt}}{\mathrm{2}}\:=\:\int_{\mathrm{0}} ^{\mathrm{2}{a}} \:\:\frac{{ln}\left({t}\right)}{\mathrm{1}−{t}^{\mathrm{2}} }{dt} \\ $$$${if}\mathrm{0}\:<\mathrm{2}{a}<\mathrm{1}\:\Leftrightarrow\:\mathrm{0}<{a}<\frac{\mathrm{1}}{\mathrm{2}} \\ $$$$\int_{\mathrm{0}} ^{\mathrm{2}{a}} \:\:\frac{{ln}\left({t}\right)}{\mathrm{1}−{t}^{\mathrm{2}} }{dt}\:=\:\int_{\mathrm{0}} ^{\mathrm{2}{a}} \left(\sum_{{n}=\mathrm{0}} ^{\infty} \:{t}^{\mathrm{2}{n}} \right){ln}\left({t}\right){dt} \\ $$$$=\:\sum_{{n}=\mathrm{0}} ^{\infty} \:\:\int_{\mathrm{0}} ^{\mathrm{2}{a}} \:{t}^{\mathrm{2}{n}} \:{ln}\left({t}\right){dt}\:\:=\sum_{{n}=\mathrm{0}} ^{\infty} \:{A}_{{n}} \\ $$$${A}_{{n}} =\int_{\mathrm{0}} ^{\mathrm{2}{a}} \:{t}^{\mathrm{2}{n}} {ln}\left({t}\right){dt}\:\:{be}\:{calculated}\:{by}\:{recurrence} \\ $$$$….{be}\:{continued}…. \\ $$$$ \\ $$