Question Number 53284 by maxmathsup by imad last updated on 20/Jan/19
$${find}\:{f}\left({x}\right)=\int_{\mathrm{0}} ^{\infty} \:\frac{{arctan}\left({xt}\right)}{\mathrm{1}+{t}^{\mathrm{2}} }{dt}\:\:\:\:{with}\:{x}\:{real}\:. \\ $$
Commented by prof Abdo imad last updated on 21/Jan/19
$${we}\:{have}\:{f}^{'} \left({x}\right)=\int_{\mathrm{0}} ^{\infty} \:\frac{{t}}{\left(\mathrm{1}+{x}^{\mathrm{2}} {t}^{\mathrm{2}} \right)\left(\mathrm{1}+{t}^{\mathrm{2}} \right)}{dt} \\ $$$$=_{{xt}\:={u}} \:\int_{\mathrm{0}} ^{\infty} \:\:\frac{{u}}{{x}\left(\mathrm{1}+{u}^{\mathrm{2}} \right)\left(\mathrm{1}+\frac{{u}^{\mathrm{2}} }{{x}^{\mathrm{2}} }\right)}\:\frac{{du}}{{x}} \\ $$$$=\int_{\mathrm{0}} ^{\infty} \:\:\:\:\frac{{u}}{\left.\left({u}^{\mathrm{2}} \:+\mathrm{1}\right)\right)\left({u}^{\mathrm{2}} \:+{x}^{\mathrm{2}} \right)}{du}\:{let}\:{decompose} \\ $$$${F}\left({u}\right)=\frac{{u}}{\left({u}^{\mathrm{2}} \:+\mathrm{1}\right)\left({u}^{\mathrm{2}} \:+{x}^{\mathrm{2}} \right)}\:\Rightarrow \\ $$$${F}\left({u}\right)=\frac{{au}+{b}}{{u}^{\mathrm{2}} \:+\mathrm{1}}\:+\frac{{cu}\:+{d}}{{u}^{\mathrm{2}} \:+{x}^{\mathrm{2}} } \\ $$$${F}\left(−{u}\right)=−{F}\left({u}\right)\:\Rightarrow\frac{−{au}\:+{b}}{{u}^{\mathrm{2}} \:+\mathrm{1}}\:+\frac{−{cu}\:+{d}}{{u}^{\mathrm{2}} \:+{x}^{\mathrm{2}} } \\ $$$$=\frac{−{au}−{b}}{{u}^{\mathrm{2}} \:+\mathrm{1}}\:+\frac{−{cu}−{d}}{{u}^{\mathrm{2}} \:+{x}^{\mathrm{2}} }\:\Rightarrow{b}={d}=\mathrm{0}\:\Rightarrow \\ $$$${F}\left({u}\right)\:=\frac{{au}}{{u}^{\mathrm{2}} \:+\mathrm{1}}\:+\frac{{cu}}{{u}^{\mathrm{2}} \:+{x}^{\mathrm{2}} } \\ $$$${lim}_{{u}\rightarrow+\infty} {u}\:{F}\left({u}\right)=\mathrm{0}\:={a}+{c}\:\Rightarrow{c}=−{a}\:\Rightarrow \\ $$$${F}\left({u}\right)\:=\frac{{au}}{{u}^{\mathrm{2}} \:+\mathrm{1}}\:−\frac{{au}}{{u}^{\mathrm{2}} \:+{x}^{\mathrm{2}} } \\ $$$${F}\left(\mathrm{1}\right)=\:\frac{\mathrm{1}}{\mathrm{2}\left(\mathrm{1}+{x}^{\mathrm{2}} \right)}\:=\frac{{a}}{\mathrm{2}}−\frac{{a}}{\mathrm{1}+{x}^{\mathrm{2}} }=\frac{{a}\:+{ax}^{\mathrm{2}} −\mathrm{2}{a}}{\mathrm{2}\left(\mathrm{1}+{x}^{\mathrm{2}} \right)}\:\Rightarrow \\ $$$$\left({x}^{\mathrm{2}} −\mathrm{1}\right){a}\:=\mathrm{1}\:\:\Rightarrow{a}\:=\frac{\mathrm{1}}{{x}^{\mathrm{2}} −\mathrm{1}}\:\left({we}\:{suppose}\:{x}\neq\overset{−} {+}\mathrm{1}\right)\:\Rightarrow \\ $$$${F}\left({u}\right)\:=\frac{\mathrm{1}}{{x}^{\mathrm{2}} −\mathrm{1}}\left\{\:\frac{{u}}{{u}^{\mathrm{2}} \:+\mathrm{1}}\:−\frac{{u}}{{u}^{\mathrm{2}} \:+{x}^{\mathrm{2}} }\right\}\:\Rightarrow \\ $$$${f}^{'} \left({x}\right)\:=\frac{\mathrm{1}}{{x}^{\mathrm{2}} −\mathrm{1}}\int_{\mathrm{0}} ^{\infty} \:\left(\frac{{u}}{{u}^{\mathrm{2}} +\mathrm{1}}\:−\frac{{u}}{{u}^{\mathrm{2}} \:+{x}^{\mathrm{2}} }\right){du} \\ $$$$=\frac{\mathrm{1}}{\mathrm{2}\left({x}^{\mathrm{2}} −\mathrm{1}\right)}\left[{ln}\mid\frac{{u}^{\mathrm{2}\:} +\mathrm{1}}{{u}^{\mathrm{2}} \:+{x}^{\mathrm{2}} }\mid\right]_{\mathrm{0}} ^{+\infty} =\frac{\mathrm{1}}{\mathrm{2}\left({x}^{\mathrm{2}} −\mathrm{1}\right)}\left(\mathrm{2}{ln}\mid{x}\mid\right) \\ $$$$=\frac{{ln}\mid{x}\mid}{{x}^{\mathrm{2}} −\mathrm{1}}\:\:{let}\:{suppose}\:{x}>\mathrm{1}\:\Rightarrow \\ $$$${f}^{'} \left({x}\right)\:=\frac{{ln}\left({x}\right)}{{x}^{\mathrm{2}} −\mathrm{1}}\:\Rightarrow\:{f}\left({x}\right)=\int_{\mathrm{1}} ^{{x}} \frac{{ln}\left({t}\right)}{{t}^{\mathrm{2}} −\mathrm{1}}\:{dt}\:+\lambda \\ $$$$\lambda\:={f}\left(\mathrm{1}\right)\:=\int_{\mathrm{0}} ^{\infty} \:\:\frac{{arctan}\left({t}\right)}{\mathrm{1}+{t}^{\mathrm{2}} }{dt}\:{by}\:{parts} \\ $$$$\lambda\:=\left[{arctan}^{\mathrm{2}} \left({t}\right)\right]_{\mathrm{0}} ^{\infty} \:−\int_{\mathrm{0}} ^{\infty} \:\frac{{arctan}\left({t}\right)}{\mathrm{1}+{t}^{\mathrm{2}} }{dt} \\ $$$$=\frac{\pi^{\mathrm{2}} }{\mathrm{4}}\:−\lambda\:\Rightarrow\mathrm{2}\lambda\:=\frac{\pi^{\mathrm{2}} }{\mathrm{4}}\:\Rightarrow\lambda=\frac{\pi^{\mathrm{2}} }{\mathrm{8}}\:\Rightarrow \\ $$$${f}\left({x}\right)=\int_{\mathrm{1}} ^{{x}} \:\:\frac{{ln}\left({t}\right)}{{t}^{\mathrm{2}} −\mathrm{1}}{dt}\:+\frac{\pi^{\mathrm{2}} }{\mathrm{8}} \\ $$
Commented by prof Abdo imad last updated on 21/Jan/19
$${let}\:{detetmine}\:\int_{\mathrm{1}} ^{{x}} \:\frac{{ln}\left({t}\right)}{{t}^{\mathrm{2}} −\mathrm{1}}{dt}\:{chsng}.{t}\:=\frac{\mathrm{1}}{{u}}\:{give} \\ $$$$\int_{\mathrm{1}} ^{{x}} \:\:\frac{{ln}\left({t}\right)}{{t}^{\mathrm{2}} −\mathrm{1}}{dt}\:=\int_{\mathrm{1}} ^{\frac{\mathrm{1}}{{x}}} \:\frac{−{ln}\left({u}\right)}{\left(\frac{\mathrm{1}}{{u}^{\mathrm{2}} }−\mathrm{1}\right)}\:\left(−\frac{{du}}{{u}^{\mathrm{2}} }\right) \\ $$$$=\int_{\mathrm{1}} ^{\frac{\mathrm{1}}{{x}}} \:\:\frac{{ln}\left({u}\right)}{{u}^{\mathrm{2}} −\mathrm{1}}\:{du}\:=\int_{\frac{\mathrm{1}}{{x}}} ^{\mathrm{1}} \:\:\frac{{ln}\left({u}\right)}{\mathrm{1}−{u}^{\mathrm{2}} }{du} \\ $$$$=\frac{\mathrm{1}}{\mathrm{2}}\int_{\frac{\mathrm{1}}{{x}}} ^{\mathrm{1}} \:{ln}\left({u}\right)\left\{\:\frac{\mathrm{1}}{\mathrm{1}−{u}}\:+\frac{\mathrm{1}}{\mathrm{1}+{u}}\right\}{du} \\ $$$$=\frac{\mathrm{1}}{\mathrm{2}}\:\int_{\frac{\mathrm{1}}{{x}}} ^{\mathrm{1}} \:\:\frac{{ln}\left({u}\right)}{\mathrm{1}−{u}}{du}\:+\frac{\mathrm{1}}{\mathrm{2}}\:\int_{\frac{\mathrm{1}}{{x}}} ^{\mathrm{1}} \:\:\frac{{ln}\left({u}\right)}{\mathrm{1}+{u}}{du} \\ $$$$\int_{\frac{\mathrm{1}}{{x}}} ^{\mathrm{1}} \:\frac{{ln}\left({u}\right)}{\mathrm{1}−{u}}{du}\:=\int_{\frac{\mathrm{1}}{{x}}} ^{\mathrm{1}} {ln}\left({u}\right)\left\{\sum_{{n}=\mathrm{0}} ^{\infty} {u}^{{n}} \right){du} \\ $$$$=\sum_{{n}=\mathrm{0}} ^{\infty} \:\int_{\frac{\mathrm{1}}{{x}}} ^{\mathrm{1}} \:{u}^{{n}} \:{ln}\left({u}\right){du}\:=\sum_{{n}=\mathrm{0}} ^{\infty} \:{A}_{{n}} \\ $$$${by}\:{parts}\:{A}_{{n}} =\left[\frac{\mathrm{1}}{{n}+\mathrm{1}}{u}^{{n}+\mathrm{1}} {ln}\left({u}\right)\right]_{\frac{\mathrm{1}}{{x}}} ^{\mathrm{1}} \:−\int_{\frac{\mathrm{1}}{{x}}} ^{\mathrm{1}} \:\frac{{u}^{{n}} }{{n}+\mathrm{1}}{du} \\ $$$$=\frac{{ln}\left({x}\right)}{\left({n}+\mathrm{1}\right){x}^{{n}+\mathrm{1}} }\:−\frac{\mathrm{1}}{\left({n}+\mathrm{1}\right)}\left[\frac{\mathrm{1}}{{n}+\mathrm{1}}\:{u}^{{n}+\mathrm{1}} \right]_{\frac{\mathrm{1}}{{x}}} ^{\mathrm{1}} \\ $$$$=\frac{{ln}\left({x}\right)}{\left({n}+\mathrm{1}\right){x}^{{n}+\mathrm{1}} }\:−\frac{\mathrm{1}}{\left({n}+\mathrm{1}\right)^{\mathrm{2}} }\left\{\mathrm{1}−\frac{\mathrm{1}}{{x}^{{n}+\mathrm{1}} }\right\}\:\Rightarrow \\ $$$$\int_{\frac{\mathrm{1}}{{x}}} ^{\mathrm{1}} \:\frac{{ln}\left({u}\right)}{\mathrm{1}−{u}}\:{du}\:={ln}\left({x}\right)\sum_{{n}=\mathrm{0}} ^{\infty} \:\frac{\mathrm{1}}{\left({n}+\mathrm{1}\right){x}^{{n}+\mathrm{1}} } \\ $$$$−\sum_{{n}=\mathrm{0}} ^{\infty} \:\frac{\mathrm{1}}{\left({n}+\mathrm{1}\right)^{\mathrm{2}} }\:+\sum_{{n}=\mathrm{0}} ^{\infty} \:\frac{\mathrm{1}}{\left({n}+\mathrm{1}\right)^{\mathrm{2}} {x}^{{n}+\mathrm{1}} } \\ $$$$\sum_{{n}=\mathrm{0}} ^{\infty} \:\frac{\mathrm{1}}{\left({n}+\mathrm{1}\right)^{\mathrm{2}} }\:=\xi\left(\mathrm{2}\right)\:=\frac{\pi^{\mathrm{2}} }{\mathrm{6}} \\ $$$${let}\:{find}\:\sum_{{n}=\mathrm{0}} ^{\infty} \:\:\frac{\mathrm{1}}{\left({n}+\mathrm{1}\right){x}^{{n}+\mathrm{1}} }\:\:{be}\:{continued}… \\ $$