Question Number 67233 by prof Abdo imad last updated on 24/Aug/19
$${calculate}\:\int_{\mathrm{0}} ^{\mathrm{1}} \:\:\:\frac{{xdx}}{\:\sqrt{\mathrm{1}+{x}^{\mathrm{4}} }} \\ $$
Commented by mind is power last updated on 24/Aug/19
$${let}\:{t}={x}^{\mathrm{2}} =>{dt}=\mathrm{2}{xdx} \\ $$$$\int_{\mathrm{0}} ^{\mathrm{1}} \frac{{dt}}{\mathrm{2}\sqrt{\left(\mathrm{1}+{t}^{\mathrm{2}} \right)}}=\frac{\mathrm{1}}{\mathrm{2}}\left[{argsh}\left({t}\right)\rceil_{\mathrm{0}} ^{\mathrm{1}} =\frac{\mathrm{1}}{\mathrm{2}}{ln}\left(\mathrm{1}+\sqrt{\mathrm{1}+\mathrm{1}^{\mathrm{2}} \:}\right)−\frac{\mathrm{1}}{\mathrm{2}}{ln}\left(\mathrm{1}+\sqrt{\mathrm{1}}\right)=\frac{\mathrm{1}}{\mathrm{2}}{ln}\left(\frac{\mathrm{3}}{\mathrm{2}}\right)\right. \\ $$