solve-integration-1-x-x-

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

s o l v e i n t e g r a t i o n \frac{1}{\sqrt{(x - α) (β - x)}} .

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

l e t u s e t h e c h . x = \frac{α - β}{2} t + \frac{α + β}{2} s o

x - α = \frac{α - β}{2} t + \frac{α + β - 2 α}{2} = \frac{α - β}{2} (t - 1) a n d

β - x = β - \frac{α + β}{2} - \frac{α - β}{2} t = \frac{β - α}{2} - \frac{α - β}{2} t

= \frac{α - β}{2} (- 1 - t) s o

I = \int ? \frac{d x}{\sqrt{(x - α) (β - x)}} = \int \frac{1}{\sqrt{\frac{α - β}{2} (t - 1) \frac{α - β}{2} (- 1 - 1 - t)}} \frac{α - β}{2} d t

= \frac{\frac{α - β}{2}}{∣ \frac{α - β}{2} ∣} \int \frac{d t}{\sqrt{1 - t^{2}}} = ξ a r c s i n (t) + k

= ξ a r c s i n (\frac{2}{α - β} (x - \frac{α + β}{2})) i f α \neq β a n d ξ^{2} = 1

a n d w e m u d t s t u d y t h e c a s e α = β \dots .