Question Number 37237 by abdo.msup.com last updated on 11/Jun/18
$${calculate}\:\:\int_{\mathrm{0}} ^{\frac{\pi}{\mathrm{2}}} \:\:\:\:\frac{{cos}\theta.{sin}\theta}{{cos}\theta\:+{sin}\theta}\:{d}\theta\:. \\ $$
Commented by abdo.msup.com last updated on 27/Jul/18
$${changement}\:{tan}\left(\frac{\theta}{\mathrm{2}}\right)={t}\:{give} \\ $$$${I}\:=\:\int_{\mathrm{0}} ^{\mathrm{1}} \:\:\:\frac{\frac{\mathrm{2}{t}\left(\mathrm{1}−{t}^{\mathrm{2}} \right)}{\left(\mathrm{1}+{t}^{\mathrm{2}} \right)^{\mathrm{2}} }}{\frac{\mathrm{1}−{t}^{\mathrm{2}} }{\mathrm{1}+{t}^{\mathrm{2}} }\:+\frac{\mathrm{2}{t}}{\mathrm{1}+{t}^{\mathrm{2}} }}\:\frac{\mathrm{2}{dt}}{\mathrm{1}+{t}^{\mathrm{2}} } \\ $$$$=\:\int_{\mathrm{0}} ^{\mathrm{1}} \:\:\:\:\:\frac{\mathrm{2}{t}\left(\mathrm{1}−{t}^{\mathrm{2}} \right)}{\left(\mathrm{1}+{t}^{\mathrm{2}} \right)^{\mathrm{2}} \left(\mathrm{1}−{t}^{\mathrm{2}} +\mathrm{2}{t}\right)}{dt} \\ $$$$=\:\int_{\mathrm{0}} ^{\mathrm{1}} \:\:\:\:\frac{\mathrm{2}{t}\left({t}^{\mathrm{2}} −\mathrm{1}\right)}{\left(\mathrm{1}+{t}^{\mathrm{2}} \right)^{\mathrm{2}} \left({t}−\mathrm{1}\right)^{\mathrm{2}} }{dt} \\ $$$$=\int_{\mathrm{0}} ^{\mathrm{1}} \:\:\:\frac{\mathrm{2}{t}\left({t}+\mathrm{1}\right)}{\left({t}^{\mathrm{2}} \:+\mathrm{1}\right)^{\mathrm{2}} \left({t}−\mathrm{1}\right)}{dt} \\ $$$$=\int_{\mathrm{0}} ^{\mathrm{1}} \:\:\:\:\frac{\mathrm{2}{t}^{\mathrm{2}} \:+\mathrm{2}{t}}{\left({t}−\mathrm{1}\right)\left({t}^{\mathrm{2}} \:+\mathrm{1}\right)^{\mathrm{2}} }{dt}\:{let}\:{decompose} \\ $$$${F}\left({t}\right)=\frac{\mathrm{2}{t}^{\mathrm{2}} +\mathrm{2}{t}}{\left({t}−\mathrm{1}\right)\left({t}^{\mathrm{2}} +\mathrm{1}\right)^{\mathrm{2}} } \\ $$$${F}\left({t}\right)=\frac{{a}}{{t}−\mathrm{1}}\:+\frac{{bt}+{c}}{{t}^{\mathrm{2}} \:+\mathrm{1}}\:+\frac{{dt}\:+{e}}{\left({t}^{\mathrm{2}} \:+\mathrm{1}\right)^{\mathrm{2}} } \\ $$$$….{be}\:{continued}… \\ $$
Answered by math1967 last updated on 11/Jun/18
$$\frac{\mathrm{1}}{\mathrm{2}}\int\frac{\mathrm{2}{sin}\theta{cos}\theta}{{cos}\theta+{sin}\theta}{d}\theta \\ $$$$\frac{\mathrm{1}}{\mathrm{2}}\int\frac{\left({sin}\theta+{cos}\theta\right)^{\mathrm{2}} }{{cos}\theta+{sin}\theta}{d}\theta\:−\frac{\mathrm{1}}{\mathrm{2}}\int\frac{{d}\theta}{{cos}\theta+{sin}\theta} \\ $$$$−\frac{\mathrm{1}}{\mathrm{2}}{cos}\theta+\frac{\mathrm{1}}{\mathrm{2}}{sin}\theta−\frac{\mathrm{1}}{\mathrm{2}\sqrt{\mathrm{2}}}\int\frac{{d}\theta}{{sin}\frac{\pi}{\mathrm{4}}{cos}\theta+{cos}\frac{\pi}{\mathrm{4}}{sin}\theta} \\ $$$$\frac{\mathrm{1}}{\mathrm{2}}\left({sin}\theta−{cos}\theta\right)−\frac{\mathrm{1}}{\mathrm{2}\sqrt{\mathrm{2}}}\int{cosec}\left(\frac{\pi}{\mathrm{4}}+\theta\right){d}\theta \\ $$$$\frac{\mathrm{1}}{\mathrm{2}}\left({sin}\theta−{cos}\theta\right)−\frac{\mathrm{1}}{\mathrm{2}\sqrt{\mathrm{2}}}{ln}\left[{cosec}\left(\frac{\pi}{\mathrm{4}}+\theta\right)−{cot}\left(\frac{\pi}{\mathrm{4}}+\theta\right)\right] \\ $$$$\therefore\underset{\mathrm{0}} {\overset{\frac{\pi}{\mathrm{2}}} {\int}}\frac{{sin}\theta{cos}\theta{d}\theta}{{sin}\theta+{cos}\theta}=\left\{\frac{\mathrm{1}}{\mathrm{2}}×\mathrm{1}−\frac{\mathrm{1}}{\mathrm{2}\sqrt{\mathrm{2}}}{ln}\left[\sqrt{\mathrm{2}}\:+\mathrm{1}\right]\right\} \\ $$$$−\left\{−\frac{\mathrm{1}}{\mathrm{2}}−\frac{\mathrm{1}}{\mathrm{2}\sqrt{\mathrm{2}}}{ln}\left[\left[\sqrt{\mathrm{2}}\:−\mathrm{1}\right]\right\}=\mathrm{1}+\frac{\mathrm{1}}{\mathrm{2}\sqrt{\mathrm{2}}}{ln}\frac{\sqrt{\mathrm{2}}\:−\mathrm{1}}{\:\sqrt{\mathrm{2}}\:+\mathrm{1}}\right. \\ $$
Commented by math1967 last updated on 11/Jun/18
$${Pls}.{check} \\ $$