# How to solve an integral.

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• Dec 15th 2011, 07:23 AM
Ad van der Ven
How to solve an integral.
How to solve the following integral (in Maple notation):

Int(Int((exp(-(u^2+v^2-2*rho*u*v)))/((1+exp(-(u^2+v^2-2*rho*u*v)))^2),u),v);

I am aware that this integral can be written as:

Int((exp(-z))/((1+exp(-z))^2),z);

with z = u^2+v^2-2*rho*u*v

I am especially interested in the following cases:

Int(Int((exp(-(u^2+v^2-2*rho*u*v)))/((1+exp(-(u^2+v^2-2*rho*u*v)))^2),u=-infinity..x),v=-infinity..y);

Int(Int(u*v*(exp(-(u^2+v^2-2*rho*u*v)))/((1+exp(-(u^2+v^2-2*rho*u*v)))^2),u=-infinity..infinity),v=-infinity..infinity);

Thanks a lot for any help.
• Dec 16th 2011, 12:50 AM
FernandoRevilla
Re: How to solve an integral.
Quote:

Originally Posted by Ad van der Ven
Int(Int((exp(-(u^2+v^2-2*rho*u*v)))/((1+exp(-(u^2+v^2-2*rho*u*v)))^2),u),v);

It is rather difficult to understand it. Perhaps you mean $\iint_{\mathbb{R}^2}\frac{e^{-(u^2+v^2-2\rho uv)}}{1+e^{-(u^2+v^2-2\rho uv)}}dudv$ . If so, and $1-\rho^2>0$ the quadratic form $Q(u,v)=u^2+v^2-2\rho uv=(u,v)\begin{pmatrix}{1}&{-\rho}\\{-\rho}&{1}\end{pmatrix}\begin{pmatrix}{u}\\{v}\end{ pmatrix}$ is positive definite and by means of the Spectral Theorem (diagonalizing $Q$) and the Euler's integral you can express the given integral in terms of product of the eigenvalues of the matrix of $Q$ (i.e. its determinant $1-\rho^2$) .