∫∫((dxdy)/(1−x^2 y^2 ))


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Question Number 122412 by mohammad17 last updated on 16/Nov/20

$\int \int \frac{d x d y}{1 - x^{2} y^{2}}$
	Answered by Olaf last updated on 16/Nov/20

	$I (x, y) = \int \int \frac{d x d y}{1 - x^{2} y^{2}}$ $I (x, y) = \int \int \frac{d x d y}{(1 - x y) (1 + x y)}$ $I (x, y) = \frac{1}{2} \int \int (\frac{1}{1 - x y} + \frac{1}{1 + x y}) d x d y$ $I (x, y) = \frac{1}{2} \int (- \frac{1}{y} \ln ∣ 1 - x y ∣ + \frac{1}{y} \ln ∣ 1 + x y ∣) d y$ $I (x, y) = - \frac{1}{2} \int \frac{1}{y} \ln (\frac{1 - x y}{1 + x y}) d y$ $Let u = x y$ $I (x, y) = - \frac{1}{2} \int \frac{x}{u} \ln (\frac{1 - u}{1 + u}) \frac{d u}{x}$ $I (x, y) = - \frac{1}{2} \int \ln (\frac{1 - u}{1 + u}) \frac{d u}{u}$ $I (x, y) = \frac{1}{2} (dilog (1 - u) - dilog (1 + u))$ $I (x, y) = \frac{1}{2} (dilog (1 - x y) - dilog (1 + x y))$
	Commented by mohammad17 last updated on 16/Nov/20

	$w h a t s t h e m e a n o f d i l o g ?$
	Commented by Olaf last updated on 17/Nov/20

	$The Spence function or dilogarithm,$ $denoted {Li}_{2} or dilog, is a special case of$ $polylogarithm . Two special functions$ $are called the Spence function :$ ${Li}_{2} (z) = - \int_{0}^{z} \frac{\ln (1 - u)}{u} d u = \int_{1}^{1 - z} \frac{\ln t}{1 - t} d t, z \in C$
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