Question Number 154669 by SANOGO last updated on 20/Sep/21
Answered by ARUNG_Brandon_MBU last updated on 20/Sep/21
$${I}=\int_{\mathrm{0}} ^{\mathrm{1}} \frac{{dt}}{\:\sqrt{\mathrm{1}+{t}^{\mathrm{2}} }+\sqrt{\mathrm{1}−{t}^{\mathrm{2}} }}=\frac{\mathrm{1}}{\mathrm{2}}\int_{\mathrm{0}} ^{\mathrm{1}} \left(\sqrt{\mathrm{1}+{t}^{\mathrm{2}} }−\sqrt{\mathrm{1}−{t}^{\mathrm{2}} }\right){dt} \\ $$$$\:\:=\frac{\mathrm{1}}{\mathrm{2}}\int_{\mathrm{0}} ^{\mathrm{argsh}\left(\mathrm{1}\right)} \mathrm{cosh}^{\mathrm{2}} \vartheta{d}\vartheta−\frac{\mathrm{1}}{\mathrm{2}}\int_{\mathrm{0}} ^{\frac{\pi}{\mathrm{2}}} \mathrm{cos}^{\mathrm{2}} \vartheta{d}\vartheta \\ $$$$\:\:=\frac{\mathrm{1}}{\mathrm{4}}\int_{\mathrm{0}} ^{\mathrm{argsh}\left(\mathrm{1}\right)} \left(\mathrm{1}+\mathrm{cosh2}\vartheta\right){d}\vartheta−\frac{\mathrm{1}}{\mathrm{4}}\int_{\mathrm{0}} ^{\frac{\pi}{\mathrm{2}}} \left(\mathrm{cos2}\vartheta+\mathrm{1}\right){d}\vartheta \\ $$$$\:\:=\frac{\mathrm{1}}{\mathrm{4}}\mathrm{argsh}\left(\mathrm{1}\right)+\left[\frac{\mathrm{sinh2}\vartheta}{\mathrm{8}}\right]_{\mathrm{0}} ^{\mathrm{argsh}\left(\mathrm{1}\right)} −\frac{\mathrm{1}}{\mathrm{4}}\left[\frac{\mathrm{sin2}\vartheta}{\mathrm{2}}+\vartheta\right]_{\mathrm{0}} ^{\frac{\pi}{\mathrm{2}}} \\ $$$$\:\:=\frac{\mathrm{1}}{\mathrm{4}}\mathrm{ln}\left(\mathrm{1}+\sqrt{\mathrm{2}}\right)+\frac{\sqrt{\mathrm{2}}}{\mathrm{4}}−\frac{\pi}{\mathrm{8}} \\ $$
Commented by ARUNG_Brandon_MBU last updated on 20/Sep/21
$$\mathrm{Ah}\:\mathrm{oui}\:!\:\mathrm{merci}. \\ $$
Commented by puissant last updated on 20/Sep/21
$${Broo}\:{attention}\:!!!! \\ $$$$\frac{\mathrm{1}}{\:\sqrt{\mathrm{1}+{t}^{\mathrm{2}} }+\sqrt{\mathrm{1}−{t}^{\mathrm{2}} }}=\frac{\sqrt{\mathrm{1}+{t}^{\mathrm{2}} }−\sqrt{\mathrm{1}−{t}^{\mathrm{2}} }}{\mathrm{2}{t}^{\mathrm{2}} }.. \\ $$
Commented by SANOGO last updated on 20/Sep/21
$${supere}\:{demonstratin} \\ $$
Answered by ARUNG_Brandon_MBU last updated on 20/Sep/21
$${I}=\int_{\mathrm{0}} ^{\mathrm{1}} \frac{{dt}}{\:\sqrt{\mathrm{1}+{t}^{\mathrm{2}} }+\sqrt{\mathrm{1}−{t}^{\mathrm{2}} }}=\int_{\mathrm{0}} ^{\mathrm{1}} \frac{\sqrt{\mathrm{1}+{t}^{\mathrm{2}} }−\sqrt{\mathrm{1}−{t}^{\mathrm{2}} }}{\mathrm{2}{t}^{\mathrm{2}} }{dt} \\ $$$$\:\:=\frac{\mathrm{1}}{\mathrm{2}}\int_{\mathrm{0}} ^{\mathrm{argsh}\left(\mathrm{1}\right)} \frac{\mathrm{cosh}^{\mathrm{2}} \vartheta}{\mathrm{sinh}^{\mathrm{2}} \vartheta}{d}\vartheta−\frac{\mathrm{1}}{\mathrm{2}}\int_{\mathrm{0}} ^{\frac{\pi}{\mathrm{2}}} \frac{\mathrm{cos}^{\mathrm{2}} \vartheta}{\mathrm{sin}^{\mathrm{2}} \vartheta}{d}\vartheta \\ $$$$\:\:=\frac{\mathrm{1}}{\mathrm{2}}\int_{\mathrm{0}} ^{\mathrm{argsh}\left(\mathrm{1}\right)} \left(\mathrm{cosech}^{\mathrm{2}} \vartheta+\mathrm{1}\right){d}\vartheta−\frac{\mathrm{1}}{\mathrm{2}}\int_{\mathrm{0}} ^{\frac{\pi}{\mathrm{2}}} \left(\mathrm{csc}^{\mathrm{2}} \vartheta−\mathrm{1}\right){d}\vartheta \\ $$$$\:\:=\frac{\mathrm{1}}{\mathrm{2}}\left[\vartheta−\mathrm{coth}\vartheta\right]_{\mathrm{0}} ^{\mathrm{argsh}\left(\mathrm{1}\right)} +\frac{\mathrm{1}}{\mathrm{2}}\left[\mathrm{cot}\vartheta+\vartheta\right]_{\mathrm{0}} ^{\frac{\pi}{\mathrm{2}}} \\ $$$$\:\:=\frac{\mathrm{1}}{\mathrm{2}}\mathrm{ln}\left(\mathrm{1}+\sqrt{\mathrm{2}}\right)−\frac{\sqrt{\mathrm{2}}}{\mathrm{2}}+\frac{\pi}{\mathrm{4}} \\ $$
Commented by SANOGO last updated on 20/Sep/21
$${merci}\:{bien}\:{mon}\:{grand} \\ $$