show that ∫_0 ^( ∞ ) ((tan^(−1) 𝛂x tan^(−1) 𝛃x)/x^2 )dx = (𝛑/2)log{(((𝛂+𝛃)^(𝛂+𝛃) )/(𝛂^𝛂 𝛃^𝛃 ))}


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Question Number 186800 by universe last updated on 10/Feb/23
$show that ∫_0 ^( ∞ ) ((tan^(−1) 𝛂x tan^(−1) 𝛃x)/x^2 )dx = (𝛑/2)log{(((𝛂+𝛃)^(𝛂+𝛃) )/(𝛂^𝛂 𝛃^𝛃 ))}$
$show that$ $\int_{0}^{\infty} \frac{\tan^{- 1} α x \tan^{- 1} β x}{x^{2}} d x = \frac{π}{2} \log {\frac{{(α + β)}^{α + β}}{α^{α} β^{β}}}$
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