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Question Number 28702 by students last updated on 29/Jan/18

solve integration (1/(√((x−α)(β−x)))) .

solveintegration1(x−α)(β−x).

Commented by abdo imad last updated on 29/Jan/18

let use the ch. x= ((α−β)/2)t +((α+β)/2)so x−α= ((α−β)/2)t +((α+β −2α)/2)=((α−β)/2) (t−1) and β−x=β−((α+β)/2) −((α−β)/2)t =((β−α)/2) −((α−β)/2)t = ((α−β)/2)(−1−t) so I=∫ ? (dx/(√((x−α)(β−x))))= ∫ (1/(√(((α−β)/2)(t−1)((α−β)/2)(−1−1−t))))((α−β)/2)dt =(((α−β)/2)/(∣((α−β)/2)∣)) ∫ (dt/(√(1−t^2 )))= ξ arcsin(t)+k = ξ arcsin ( (2/(α−β))(x− ((α+β)/2))) if α≠β and ξ^2 =1 and we mudt study the case α=β....

letusethech.x=α−β2t+α+β2sox−α=α−β2t+α+β−2α2=α−β2(t−1)andβ−x=β−α+β2−α−β2t=β−α2−α−β2t=α−β2(−1−t)soI=∫?dx(x−α)(β−x)=∫1α−β2(t−1)α−β2(−1−1−t)α−β2dt=α−β2∣α−β2∣∫dt1−t2=ξarcsin(t)+k=ξarcsin(2α−β(x−α+β2))ifα≠βandξ2=1andwemudtstudythecaseα=β....

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