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Proof that the integral from a to b of 1/√(x-a)(b-x) dx =π by three method

\int_{a}^{b} \frac{1}{\sqrt{(x-a)(b-x)}}dx

We will Proof that  ba1(xa)(bx)dx=π  by three method :

The first method:

let: x-a=(b-a)y  ⇒ x(b-a)y + a ⇒  dx=(b-a)dy

     b-x=b-(b-a)y-a ⇒ b-x=(b-a)(1-y)

     x=a  ⇒    y=o      ,     x=b     ⇒     y=1

\int_{a}^{b} \frac{1}{\sqrt{(x-a)(b-x)}}dx



The second method:

 let: x =a cos²Ө+b sin²Ө ⇒  x-a= (b-a) sin²Ө ⇒  b-x=(b-a) cos²Ө

 dx-2a sinӨ . cosӨ dӨ + 2b sinӨ . cosӨ dӨ = (b-a)sin 2Ө dӨ

                x=b     ⇒     Ө=π/2    ,    x=a   ⇒     Ө =0

Integration math


The third method:

let:   x-a=y    ⇒   x=y+a   ⇒   dx=dy
        x=b     ⇒  y=b-a    ,    x=a  ⇒     y=0




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