dx-x-x-x-pi-where-gt-0-

Question Number 151341 by peter frank last updated on 20/Aug/21

\int_{α}^{β} \frac{dx}{x \sqrt{(x - α) (β - x)}} = \frac{π}{\sqrt{α β}}

where α, β > 0

Answered by Kamel last updated on 20/Aug/21

Ω (α, β) = \int_{α}^{β} \frac{d x}{x \sqrt{(x - α) (β - x)}}, β > α > 0 .

(x - α) z = \sqrt{(x - α) (β - x)}

(x - α) z^{2} = (β - x) \Rightarrow x = \frac{β + α z^{2}}{1 + z^{2}}, d x = - 2 z (β - α) \frac{d z}{{(1 + z^{2})}^{2}}

x - α = \frac{β + α z^{2}}{1 + z^{2}} - α = \frac{β - α}{1 + z^{2}}

∴ Ω (α, β) = 2 \int_{0}^{+ \infty} \frac{d z}{(β + α z^{2})} = \frac{π}{\sqrt{α β}}

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