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Let $A=[0,1]\times[0,1]$ and $f(x,y) = \frac{x}{(1+x^2)(1+xy)}$. Show that $2\int_A f(x,y)dxdy = \int_A (f(x,y) + f(y,x))dxdy$.

How can I prove it? I just obvserved that $A$ is symmetric over $y=x$, but also $f(x,y) \neq f(y,x)$, so I'm not really sure how to approach this problem.

Javier321
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  • Hint: $\displaystyle \int_0^1 \left( \int_0^1 f(x,y) \ \mathrm{d}x \right)\ \mathrm{d}y = \int_0^1 \left( \int_0^1 f(y,x) \ \mathrm{d}y \right)\ \mathrm{d}x$. – sudeep5221 Mar 11 '23 at 15:54
  • @sudeep5221 Is that by just renaming $x$ to $y$ and $y$ to $x$? Or are you using a theorem for that? – Javier321 Mar 11 '23 at 16:20

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