Question Number 206892 by mathzup last updated on 29/Apr/24
$${find}\:\int_{\mathrm{0}} ^{\mathrm{1}} \sqrt{\mathrm{1}+\sqrt{\mathrm{1}+{x}^{\mathrm{2}} }}{dx} \\ $$
Answered by Frix last updated on 29/Apr/24
$${t}={x}+\sqrt{{x}^{\mathrm{2}} +\mathrm{1}} \\ $$$$\Rightarrow \\ $$$$\frac{\sqrt{\mathrm{2}}}{\mathrm{4}}\int\left({t}^{\frac{\mathrm{1}}{\mathrm{2}}} +{t}^{−\frac{\mathrm{1}}{\mathrm{2}}} +{t}^{−\frac{\mathrm{3}}{\mathrm{2}}} +{t}^{−\frac{\mathrm{5}}{\mathrm{2}}} \right){dt} \\ $$$$\mathrm{Which}\:\mathrm{is}\:\mathrm{easy} \\ $$