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




Question Number 36421 by abdo.msup.com last updated on 01/Jun/18
find ∫    (dx/(1+2(√(1−x))))
$${find}\:\int\:\:\:\:\frac{{dx}}{\mathrm{1}+\mathrm{2}\sqrt{\mathrm{1}−{x}}} \\ $$
Commented by abdo.msup.com last updated on 04/Jun/18
changement (√(1−x))=t give 1−x=t^2   I = ∫     ((−2tdt)/(1+2t)) = −∫ ((2t+1−1)/(2t +1))dt  =−t + ∫    (dt/(2t +1)) = −t  +(1/2)ln∣2t+1∣ +c
$${changement}\:\sqrt{\mathrm{1}−{x}}={t}\:{give}\:\mathrm{1}−{x}={t}^{\mathrm{2}} \\ $$$${I}\:=\:\int\:\:\:\:\:\frac{−\mathrm{2}{tdt}}{\mathrm{1}+\mathrm{2}{t}}\:=\:−\int\:\frac{\mathrm{2}{t}+\mathrm{1}−\mathrm{1}}{\mathrm{2}{t}\:+\mathrm{1}}{dt} \\ $$$$=−{t}\:+\:\int\:\:\:\:\frac{{dt}}{\mathrm{2}{t}\:+\mathrm{1}}\:=\:−{t}\:\:+\frac{\mathrm{1}}{\mathrm{2}}{ln}\mid\mathrm{2}{t}+\mathrm{1}\mid\:+{c} \\ $$

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