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find-0-1-1-1-x-2-dx-




Question Number 206892 by mathzup last updated on 29/Apr/24
find ∫_0 ^1 (√(1+(√(1+x^2 ))))dx
$${find}\:\int_{\mathrm{0}} ^{\mathrm{1}} \sqrt{\mathrm{1}+\sqrt{\mathrm{1}+{x}^{\mathrm{2}} }}{dx} \\ $$
Answered by Frix last updated on 29/Apr/24
t=x+(√(x^2 +1))  ⇒  ((√2)/4)∫(t^(1/2) +t^(−(1/2)) +t^(−(3/2)) +t^(−(5/2)) )dt  Which is easy
$${t}={x}+\sqrt{{x}^{\mathrm{2}} +\mathrm{1}} \\ $$$$\Rightarrow \\ $$$$\frac{\sqrt{\mathrm{2}}}{\mathrm{4}}\int\left({t}^{\frac{\mathrm{1}}{\mathrm{2}}} +{t}^{−\frac{\mathrm{1}}{\mathrm{2}}} +{t}^{−\frac{\mathrm{3}}{\mathrm{2}}} +{t}^{−\frac{\mathrm{5}}{\mathrm{2}}} \right){dt} \\ $$$$\mathrm{Which}\:\mathrm{is}\:\mathrm{easy} \\ $$

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