Question Number 141916 by mnjuly1970 last updated on 24/May/21
$$ \\ $$$$\:\:\:\:\:{prove}\:\:{that}:: \\ $$$$\:\:\:\mathrm{I}:=\int_{\mathrm{0}} ^{\:\frac{\pi}{\mathrm{2}}} {arccosh}\left({sin}\left({x}\right)+{cos}\left({x}\right)\right){dx}=\frac{\pi}{\mathrm{2}}{ln}\left(\mathrm{2}\right)\:.. \\ $$
Answered by mindispower last updated on 25/May/21
$${by}\:{part}\:{withe}\:{d}\left({arccoh}\left({x}\right)\right)=\frac{{dx}}{\:\sqrt{\mathrm{1}−{x}^{\mathrm{2}} }} \\ $$$$\left.{I}={xarccosh}\left({sinx}+{cos}\left({x}\right)\right)\right]_{\mathrm{0}} ^{\frac{\pi}{\mathrm{2}}} −\int_{\mathrm{0}} ^{\frac{\pi}{\mathrm{2}}} \frac{{x}\left({cos}\left({x}\right)−{sin}\left({x}\right)\right)}{\:\sqrt{{sin}\left(\mathrm{2}{x}\right)}}{dx} \\ $$$$−\int_{\mathrm{0}} ^{\frac{\pi}{\mathrm{2}}} \frac{{xcos}\left({x}\right)−{xsin}\left({x}\right)}{\:\sqrt{{sin}\left(\mathrm{2}{x}\right)}}{dx} \\ $$$$=−\frac{\mathrm{1}}{\:\sqrt{\mathrm{2}}}\int_{\mathrm{0}} ^{\frac{\pi}{\mathrm{2}}} {xsin}^{−\frac{\mathrm{1}}{\mathrm{2}}} \left({x}\right){cos}^{\frac{\mathrm{1}}{\mathrm{2}}} \left({x}\right)+\frac{\mathrm{1}}{\:\sqrt{\mathrm{2}}}\int_{\mathrm{0}} ^{\frac{\pi}{\mathrm{2}}} {xsin}^{\frac{\mathrm{1}}{\mathrm{2}}} \left({x}\right){cos}^{−\frac{\mathrm{1}}{\mathrm{2}}} \left({x}\right) \\ $$$$\frac{\mathrm{1}}{\:\sqrt{\mathrm{2}}}\int_{\mathrm{0}} ^{\infty} \frac{{arctg}\left({x}\right)\left(\sqrt{{x}}−\frac{\mathrm{1}}{\:\sqrt{{x}}}\right)}{\mathrm{1}+{x}^{\mathrm{2}} }{dx}={A} \\ $$$$\left.\int_{\mathrm{0}} ^{\infty} \frac{{x}^{{s}} {arctan}\left({tx}\right)}{\mathrm{1}+{x}^{\mathrm{2}} }\:{dx}={f}\left({t}\right),{s}\in\right]−\mathrm{1},\mathrm{1}\left[\right. \\ $$$${f}'\left({t}\right)=\int_{\mathrm{0}} ^{\infty} \frac{{x}^{{s}+\mathrm{1}} }{\left(\mathrm{1}+{x}^{\mathrm{2}} \right)\left(\mathrm{1}+{t}^{\mathrm{2}} {x}^{\mathrm{2}} \right)}{dx} \\ $$$$\int_{−\infty} ^{\infty} \frac{{x}^{{s}+\mathrm{1}} }{\left(\mathrm{1}+{x}^{\mathrm{2}} \right)\left(\mathrm{1}+{t}^{\mathrm{2}} {x}^{\mathrm{2}} \right)}{dx}={a}=\mathrm{2}{i}\pi\left(.\frac{{i}^{{s}+\mathrm{1}} }{\mathrm{2}{i}\left(\mathrm{1}−{t}^{\mathrm{2}} \right)}+\frac{{i}^{{s}+\mathrm{1}} {t}^{\mathrm{2}} }{{t}^{{s}+\mathrm{1}} .\left({t}^{\mathrm{2}} −\mathrm{1}\right)}.\frac{\mathrm{1}}{\mathrm{2}{it}}\right) \\ $$$$=\frac{\pi{i}^{{s}+\mathrm{1}} }{\mathrm{1}−{t}^{\mathrm{2}} }\left(\mathrm{1}−{t}^{−{s}} \right) \\ $$$${a}={f}\left({t}\right)+\int_{\mathrm{0}} ^{\infty} \frac{\left(−{x}\right)^{{s}+\mathrm{1}} }{\left(\mathrm{1}+{x}^{\mathrm{2}} \right)\left(\mathrm{1}+{t}^{\mathrm{2}} {x}^{\mathrm{2}} \right)}{dx} \\ $$$$=\left(\mathrm{1}+\left(−\mathrm{1}\right)^{{s}+\mathrm{1}} \right){f}\left({t}\right) \\ $$$$=\left(\mathrm{1}+{e}^{{i}\pi\left({s}+\mathrm{1}\right)} \right){f}\left({t}\right) \\ $$$${f}\left({t}\right)=\frac{\pi}{\mathrm{1}−{t}^{\mathrm{2}} }\left(\mathrm{1}−{t}^{−{s}} \right).\frac{{e}^{{i}\frac{\pi}{\mathrm{2}}\left({s}+\mathrm{1}\right)} }{\mathrm{1}+{e}^{{i}\pi\left({s}+\mathrm{1}\right)} } \\ $$$$=\frac{\pi\left(\mathrm{1}−{t}^{−{s}} \right)}{\mathrm{2}\left(\mathrm{1}−{t}^{\mathrm{2}} \right){cos}\left(\frac{\pi}{\mathrm{2}}\left({s}+\mathrm{1}\right)\right)}={f}'\left({t}\right) \\ $$$$\int_{\mathrm{0}} ^{\mathrm{1}} {f}'\left({t}\right)=\frac{{x}^{{s}} {arctan}\left({x}\right)}{\mathrm{1}+{x}^{\mathrm{2}} } \\ $$$$=\frac{\pi}{\mathrm{2}{cos}\left(\frac{\pi}{\mathrm{2}}\left({s}+\mathrm{1}\right)\right)}\int_{\mathrm{0}} ^{\mathrm{1}} \frac{\mathrm{1}−{t}^{−{s}} }{\mathrm{1}−{t}^{\mathrm{2}} }{dt}_{{t}^{\mathrm{2}} ={x}} \\ $$$$=−\frac{\pi}{\mathrm{4}{sin}\left(\frac{\pi{s}}{\mathrm{2}}\right)}\int_{\mathrm{0}} ^{\mathrm{1}} \frac{\mathrm{1}−{x}^{−\frac{{s}}{\mathrm{2}}} }{\mathrm{1}−{x}}{x}^{−\frac{\mathrm{1}}{\mathrm{2}}} \\ $$$$=−\frac{\pi}{\mathrm{4}{sin}\left(\frac{\pi{s}}{\mathrm{2}}\right)}\int_{\mathrm{0}} ^{\mathrm{1}} \frac{{x}^{−\frac{\mathrm{1}}{\mathrm{2}}} −{x}^{−\frac{{s}+\mathrm{1}}{\mathrm{2}}} }{\mathrm{1}−{x}}{dx}={g}\left({s}\right) \\ $$$$\Psi\left({s}+\mathrm{1}\right)=−\gamma+\int_{\mathrm{0}} ^{\mathrm{1}} \frac{\mathrm{1}−{x}^{{s}} }{\mathrm{1}−{x}}{dx} \\ $$$${g}\left({s}\right)=−\frac{\pi}{\mathrm{4}{sin}\left(\frac{\pi{s}}{\mathrm{2}}\right)}.\left(\Psi\left(\frac{\mathrm{1}−{s}}{\mathrm{2}}\right)−\Psi\left(\frac{\mathrm{1}}{\mathrm{2}}\right)\right) \\ $$$${A}=−\frac{\pi}{\mathrm{4}{sin}\left(\frac{\pi}{\mathrm{4}}\right)\sqrt{\mathrm{2}}}\left(\Psi\left(\frac{\mathrm{1}}{\mathrm{4}}\right)−\mathrm{2}\Psi\left(\frac{\mathrm{1}}{\mathrm{2}}\right)+\Psi\left(\frac{\mathrm{3}}{\mathrm{4}}\right)\right) \\ $$$$\Psi\left(\frac{\mathrm{1}}{\mathrm{2}}\right)=−\mathrm{2}{ln}\left(\mathrm{2}\right)−\gamma \\ $$$$\Psi\left(\frac{\mathrm{1}}{\mathrm{4}}\right)=\left(−\frac{\pi}{\mathrm{2}}−\mathrm{3}{ln}\left(\mathrm{2}\right)−\gamma\right) \\ $$$$\Psi\left(\frac{\mathrm{3}}{\mathrm{4}}\right)=\Psi\left(\frac{\mathrm{1}}{\mathrm{4}}\right)+\pi \\ $$$$=−\frac{\pi}{\mathrm{4}}\left(−\pi−\mathrm{6}{ln}\left(\mathrm{2}\right)−\mathrm{2}\gamma+\pi+\mathrm{4}{ln}\left(\mathrm{2}\right)+\mathrm{2}\gamma\right) \\ $$$$=−\frac{\pi}{\mathrm{4}}.−\mathrm{2}{ln}\left(\mathrm{2}\right)=\frac{\pi}{\mathrm{2}}{ln}\left(\mathrm{2}\right) \\ $$$$ \\ $$$$ \\ $$$$ \\ $$$$ \\ $$$$ \\ $$$$ \\ $$$$ \\ $$$$ \\ $$$$ \\ $$
Commented by mnjuly1970 last updated on 25/May/21
$${very}\:{nice}\:.{thanks}\:{alot}… \\ $$
Commented by mindispower last updated on 25/May/21
$${withe}\:{pleasur}\:{sir} \\ $$