(adsbygoogle = window.adsbygoogle || []).push({}); 1. The problem statement, all variables and given/known data

By transforming to polar coordinates, show that

[tex]I = \int\int_{T}\frac{1}{(1+x^2)(1+y^2)}dxdy = \int^{\pi/4}_{0}\frac{log(\sqrt{2}cos(\theta))}{cos(2\theta)}d\theta[/tex]

where T is the triangle with successive vertices (0,0),(1,0),(1,1).

2. Relevant equations

[tex]I = \int\int_{K} f(x,y)dxdy = \int\int_{K'} g(u,v)*J*dudv[/tex]

where J is the Jacobian.

3. The attempt at a solution

The Jacobian is r, as always with a transformation to polar coordinates, so we get that

[tex]I = \int^{1}_{0}\int^{x}_{0}\frac{1}{(1+x^2)(1+y^2)}dydx = \int^{\pi/4}_{0}\int^{\sqrt2}_{1}\frac{r}{(1+r^2cos^2(\theta))(1+r^2sin^2(\theta))}drd\theta[/tex]

Firstly, is this correct? Secondly, if it is, could you give me a hint as to how to solve it to get the answer given? The obvious thing seems to be to split it into partial fractions, but I did try this once and didn't seem to get anywhere?

Thanks!

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# Homework Help: Multiple integral problem

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