Question Number 41913 by math khazana by abdo last updated on 15/Aug/18
$${let}\:{f}\left({a}\right)\:=\:\int_{\mathrm{0}} ^{\pi} \:\:\:\:\frac{{x}}{\mathrm{1}+{acosx}}{dx} \\ $$$$\left.\mathrm{1}\right)\:{find}\:\:{f}\left({a}\right)\: \\ $$$$\left.\mathrm{2}\right)\:{calculate}\:\:\int_{\mathrm{0}} ^{\pi} \:\:\:\frac{{x}}{\mathrm{1}+\mathrm{2}{cosx}}{dx}\:{and}\:\int_{\mathrm{0}} ^{\pi} \:\:\:\frac{{x}}{\mathrm{1}−\mathrm{2}{cosx}}{dx} \\ $$$$\left.\mathrm{3}\right)\:{calculate}\:\:\int_{\mathrm{0}} ^{\pi} \:\:\frac{{xcosx}}{\left(\mathrm{1}+{acosx}\right)^{\mathrm{2}} }{dx} \\ $$$$\left.\mathrm{4}\right)\:{find}\:{the}\:{value}\:{of}\:\:\int_{\mathrm{0}} ^{\pi} \:\:\:\frac{{xcosx}}{\left(\mathrm{1}+\mathrm{2}{cosx}\right)^{\mathrm{2}} }{dx}\:. \\ $$
Answered by maxmathsup by imad last updated on 16/Aug/18
$$\left.\mathrm{1}\right)\:{we}\:{have}\:{f}\left({a}\right)\:=\:\int_{\mathrm{0}} ^{\pi} \:\:\:\frac{{x}}{\mathrm{1}+{a}\frac{\mathrm{2}{tan}\left(\frac{{x}}{\mathrm{2}}\right)}{\mathrm{1}+{tan}^{\mathrm{2}} \left(\frac{{x}}{\mathrm{2}}\right)}}\:{dx}\:\:{changement}\:{tan}\left(\frac{{x}}{\mathrm{2}}\right)\:={t}\:{give} \\ $$$${f}\left({a}\right)\:=\int_{\mathrm{0}} ^{\infty} \:\:\:\frac{\mathrm{2}{arctan}\left({t}\right)}{\mathrm{1}+\frac{\mathrm{2}{at}}{\mathrm{1}+{t}^{\mathrm{2}} }}\:\frac{\mathrm{2}{dt}}{\mathrm{1}+{t}^{\mathrm{2}} }\:=\int_{\mathrm{0}} ^{\infty} \:\:\:\frac{\mathrm{4}{arctan}\left({t}\right)}{\mathrm{1}+{t}^{\mathrm{2}} \:+\mathrm{2}{at}}{dt}\:=\int_{\mathrm{0}} ^{\infty} \:\:\:\frac{\mathrm{4}{arctan}\left({t}\right)}{{t}^{\mathrm{2}} \:+\mathrm{2}{at}\:+\mathrm{1}}\:{dt} \\ $$$$=\mathrm{4}\:\int_{\mathrm{0}} ^{\infty} \:\:\:\:\:\frac{{arctan}\left({t}\right)}{{t}^{\mathrm{2}} \:+\mathrm{2}{at}\:+\mathrm{1}}{dt}\:={w}\left(\mathrm{1}\right)\:{with}\:{w}\left(\alpha\right)\:=\int_{\mathrm{0}} ^{\infty} \:\:\:\:\frac{{arctan}\left(\alpha{t}\right)}{{t}^{\mathrm{2}} \:+\mathrm{2}{at}\:+\mathrm{1}}\:{dt}\:{we}\:{have} \\ $$$${w}^{'} \left(\alpha\right)\:=\int_{\mathrm{0}} ^{\infty} \:\:\:\:\:\:\:\frac{{t}}{\left(\mathrm{1}+\alpha^{\mathrm{2}} {t}^{\mathrm{2}} \right)\left({t}^{\mathrm{2}} \:+\mathrm{2}{at}\:+\mathrm{1}\right)}\:{dt}\:\:=_{\alpha{t}\:={u}} \:\:\int_{\mathrm{0}} ^{\infty} \:\:\:\:\:\:\:\frac{{u}}{\alpha\left(\mathrm{1}+{u}^{\mathrm{2}} \right)\left(\frac{{u}^{\mathrm{2}} }{\alpha^{\mathrm{2}} }\:+\mathrm{2}{a}\frac{{u}}{\alpha}+\mathrm{1}\right)}\frac{{du}}{\alpha} \\ $$$$=\:\int_{\mathrm{0}} ^{\infty} \:\:\:\:\:\:\:\:\frac{{udu}}{\alpha^{\mathrm{2}} \left(\mathrm{1}+{u}^{\mathrm{2}} \right)\left(\frac{{u}^{\mathrm{2}} \:+\mathrm{2}\alpha{au}\:+\alpha^{\mathrm{2}} }{\alpha^{\mathrm{2}} }\right)}\:=\int_{\mathrm{0}} ^{\infty} \:\:\:\:\:\frac{{udu}}{\left({u}^{\mathrm{2}} \:+\mathrm{1}\right)\left({u}^{\mathrm{2}} \:+\mathrm{2}\alpha{au}\:+\alpha^{\mathrm{2}} \right)} \\ $$$${let}\:{decompose}\:{F}\left({u}\right)\:=\:\frac{{u}}{\left({u}^{\mathrm{2}} \:+\mathrm{1}\right)\left({u}^{\mathrm{2}} \:+\mathrm{2}\alpha{au}\:+\alpha^{\mathrm{2}} \right)}\:\left({let}\:{take}\:\alpha>\mathrm{0}\right) \\ $$$${roots}\:{of}\:\:{u}^{\mathrm{2}} \:+\mathrm{2}\alpha{au}\:+\alpha^{\mathrm{2}} \\ $$$$\Delta^{'} \:=\alpha^{\mathrm{2}} {a}^{\mathrm{2}} \:−\alpha^{\mathrm{2}} \:=\alpha^{\mathrm{2}} \left({a}^{\mathrm{2}} −\mathrm{1}\right) \\ $$$$\left.{case}\mathrm{1}\right)\:\mid{a}\mid>\mathrm{1}\:\:\Rightarrow\:{u}_{\mathrm{1}} =−\alpha{a}\:+\mid\alpha\mid\sqrt{{a}^{\mathrm{2}} \:−\mathrm{1}}\:\:\:{and}\:\:{u}_{\mathrm{2}} =−\alpha{a}\:−\mid\alpha\mid\sqrt{{a}^{\mathrm{2}} \:−\mathrm{1}} \\ $$$${F}\left({u}\right)\:=\:\:\frac{{a}}{{u}−{u}_{\mathrm{1}} }\:+\frac{{b}}{{u}−{u}_{\mathrm{2}} }\:\:+\frac{{cu}\:+{d}}{{u}^{\mathrm{2}} \:+\mathrm{1}}\:\:=\frac{{u}}{\left({u}−{u}_{\mathrm{1}} \right)\left({u}−{u}_{\mathrm{2}} \right)\left({u}^{\mathrm{2}} \:+\mathrm{1}\right)} \\ $$$${a}\:=\:\:\frac{{u}_{\mathrm{1}} }{\left({u}_{\mathrm{1}} −{u}_{\mathrm{2}} \right)\left({u}_{\mathrm{1}} ^{\mathrm{2}} \:+\mathrm{1}\right)}\:=\frac{\mid\alpha\mid\sqrt{{a}^{\mathrm{2}} −\mathrm{1}}−\alpha{a}}{\mathrm{2}\mid\alpha\mid\sqrt{{a}^{\mathrm{2}} −\mathrm{1}}\left(\left(\mid\alpha\mid\sqrt{{a}^{\mathrm{2}} −\mathrm{1}}−\alpha{a}\right)^{\mathrm{2}} +\mathrm{1}\right)}\:=\frac{\alpha\left(\sqrt{{a}^{\mathrm{2}} −\mathrm{1}}−{a}\right)}{\mathrm{2}\alpha\sqrt{{a}^{\mathrm{2}} −\mathrm{1}}\left(\alpha^{\mathrm{2}} \left(\sqrt{{a}^{\mathrm{2}} −\mathrm{1}}−{a}\right)^{\mathrm{2}} +\mathrm{1}\right)} \\ $$$$=\frac{\sqrt{{a}^{\mathrm{2}} −\mathrm{1}}−{a}}{\mathrm{2}\sqrt{{a}^{\mathrm{2}} −\mathrm{1}}\left\{\alpha^{\mathrm{2}} \left(\sqrt{{a}^{\mathrm{2}} −\mathrm{1}}−{a}\right)^{\mathrm{2}} \:+\mathrm{1}\right\}} \\ $$$${b}\:=\:\frac{{u}_{\mathrm{2}} }{\left({u}_{\mathrm{2}} −{u}_{\mathrm{1}} \right)\left({u}_{\mathrm{2}} ^{\mathrm{2}} \:+\mathrm{1}\right)}\:=\:\frac{−\alpha\left({a}+\sqrt{{a}^{\mathrm{2}} −\mathrm{1}}\right)}{−\mathrm{2}\alpha\sqrt{{a}^{\mathrm{2}} −\mathrm{1}}\left\{\:\alpha^{\mathrm{2}} \left({a}+\sqrt{{a}^{\mathrm{2}} −\mathrm{1}}\right)^{\mathrm{2}} \:+\mathrm{1}\right\}} \\ $$$${lim}_{{u}\rightarrow+\infty} {u}\:{F}\left({u}\right)\:=\mathrm{0}\:={a}+{b}\:+{c}\:\Rightarrow\:{c}\:=−{a}−{b}\:\Rightarrow \\ $$$${F}\left({u}\right)\:=\:\frac{{a}}{{u}−{u}_{\mathrm{1}} }\:+\frac{{b}}{{u}−{u}_{\mathrm{2}} }\:+\frac{\left(−{a}−{b}\right){u}\:+{d}}{{u}^{\mathrm{2}} \:+\mathrm{1}} \\ $$$${F}\left(\mathrm{0}\right)\:=\mathrm{0}\:=−\frac{{a}}{{u}_{\mathrm{1}} }\:−\frac{{b}}{{u}_{\mathrm{2}} }\:+{d}\:\Rightarrow\:{d}\:=\frac{{a}}{{u}_{\mathrm{1}} }\:+\frac{{b}}{{u}_{\mathrm{2}} }\:\Rightarrow \\ $$$${F}\left({u}\right)\:=\:\frac{{a}}{{u}−{u}_{\mathrm{1}} }\:+\frac{{b}}{{u}−{u}_{\mathrm{2}} }\:+\frac{\left(−{a}−{b}\right){u}\:\:+\frac{{a}}{{u}_{\mathrm{1}} }+\frac{{b}}{{u}_{\mathrm{2}} }}{{u}^{\mathrm{2}} \:+\mathrm{1}}\:\Rightarrow \\ $$$$\int_{\mathrm{0}} ^{\infty} \:\:{F}\left({u}\right){du}\:\:=\left[{aln}\mid{u}−{u}_{\mathrm{1}} \mid\:+{bln}\mid{u}−{u}_{\mathrm{2}} \mid\right]_{\mathrm{0}} ^{+\infty} \:\:−\left[\frac{{a}+{b}}{\mathrm{2}}{ln}\left({u}^{\mathrm{2}} \:+\mathrm{1}\right)\right]_{\mathrm{0}} ^{+\infty} \\ $$$$+\left(\frac{{a}}{{u}_{\mathrm{1}} }\:+\frac{{b}}{{u}_{\mathrm{2}} }\right)\left[\:{arctan}\left({u}\right)\right]_{\mathrm{0}} ^{+\infty} \\ $$$$=\left[\:{ln}\left(\mid{u}−{u}_{\mathrm{1}} \mid^{{a}} \mid{u}−{u}_{\mathrm{2}} \mid^{{b}} \right)−{ln}\left(\left({u}^{\mathrm{2}} \:+\mathrm{1}\right)^{\frac{{a}+{b}}{\mathrm{2}}} \right)\right]_{\mathrm{0}} ^{+\infty} \:+\frac{\pi}{\mathrm{2}}\left\{\frac{{a}}{{u}_{\mathrm{1}} }\:+\frac{{b}}{{u}_{\mathrm{2}} }\right\}\:\:…{be}\:{continued}… \\ $$$$ \\ $$