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calculate-0-1-xdx-1-x-4-




Question Number 67233 by prof Abdo imad last updated on 24/Aug/19
calculate ∫_0 ^1    ((xdx)/( (√(1+x^4 ))))
$${calculate}\:\int_{\mathrm{0}} ^{\mathrm{1}} \:\:\:\frac{{xdx}}{\:\sqrt{\mathrm{1}+{x}^{\mathrm{4}} }} \\ $$
Commented by mind is power last updated on 24/Aug/19
let t=x^2 =>dt=2xdx  ∫_0 ^1 (dt/(2(√((1+t^2 )))))=(1/2)[argsh(t)⌉_0 ^1 =(1/2)ln(1+(√(1+1^2  )))−(1/2)ln(1+(√1))=(1/2)ln((3/2))
$${let}\:{t}={x}^{\mathrm{2}} =>{dt}=\mathrm{2}{xdx} \\ $$$$\int_{\mathrm{0}} ^{\mathrm{1}} \frac{{dt}}{\mathrm{2}\sqrt{\left(\mathrm{1}+{t}^{\mathrm{2}} \right)}}=\frac{\mathrm{1}}{\mathrm{2}}\left[{argsh}\left({t}\right)\rceil_{\mathrm{0}} ^{\mathrm{1}} =\frac{\mathrm{1}}{\mathrm{2}}{ln}\left(\mathrm{1}+\sqrt{\mathrm{1}+\mathrm{1}^{\mathrm{2}} \:}\right)−\frac{\mathrm{1}}{\mathrm{2}}{ln}\left(\mathrm{1}+\sqrt{\mathrm{1}}\right)=\frac{\mathrm{1}}{\mathrm{2}}{ln}\left(\frac{\mathrm{3}}{\mathrm{2}}\right)\right. \\ $$

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