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x-1-2x-x-2-dx-




Question Number 59807 by aliesam last updated on 15/May/19
∫((x−1)/( (√(2x−x^2 )))) dx
$$\int\frac{{x}−\mathrm{1}}{\:\sqrt{\mathrm{2}{x}−{x}^{\mathrm{2}} }}\:{dx} \\ $$
Answered by tanmay last updated on 15/May/19
t^2 =2x−x^2   2tdt=(2−2x)dx  tdt=(1−x)dx  −tdt=(x−1)dx  ∫((−tdt)/t)  =−t+c  =(−1)(2x−x^2 )^(1/2) +c
$${t}^{\mathrm{2}} =\mathrm{2}{x}−{x}^{\mathrm{2}} \\ $$$$\mathrm{2}{tdt}=\left(\mathrm{2}−\mathrm{2}{x}\right){dx} \\ $$$${tdt}=\left(\mathrm{1}−{x}\right){dx} \\ $$$$−{tdt}=\left({x}−\mathrm{1}\right){dx} \\ $$$$\int\frac{−{tdt}}{{t}} \\ $$$$=−{t}+{c} \\ $$$$=\left(−\mathrm{1}\right)\left(\mathrm{2}{x}−{x}^{\mathrm{2}} \right)^{\frac{\mathrm{1}}{\mathrm{2}}} +{c} \\ $$

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