prove-that-0-100-dx-x-100-x-pi-

Question Number 127904 by bramlexs22 last updated on 03/Jan/21

prove that \int_{0}^{100} \frac{dx}{\sqrt{x (100 - x)}} = π

Answered by liberty last updated on 03/Jan/21

let x = 100 \sin^{2} t \Rightarrow dx = 200 \sin t \cos t dt

\int_{0}^{\frac{π}{2}} \frac{200 \sin t \cos t dt}{\sqrt{100 \sin^{2} t (100 \cos^{2} t})} =

{\int_{0}^{\frac{π}{2}} 2 dt = 2 t]}_{0}^{\frac{π}{2}} = 2 \times \frac{π}{2} = π

Answered by Dwaipayan Shikari last updated on 03/Jan/21

\int_{0}^{100} \frac{d x}{\sqrt{x} \sqrt{100 - x}} x = 100 u

= \int_{0}^{1} \frac{d u}{\sqrt{u} \sqrt{1 - u}} = β (\frac{1}{2}, \frac{1}{2}) = Γ^{2} (\frac{1}{2}) = \sqrt{π} . \sqrt{π} = π