How can I find this integral $$I=\int_{-4}^4\int_{-3}^3 \int_{-2}^2 \int_{-1}^1 \frac{x_1-x_2+x_3{-}x_4}{x_1+x_2+x_3+x_4} \, dx_1 \, dx_2 \, dx_3 \, dx_4$$
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$\begingroup$ Maple says undefined, calculating it numerically. Have you tried a change of variables $$u=x_1+x_2+x_3+x_4, v=x_1-x_2-x_3+x_4, w=x_3,z=x_4?$$ $\endgroup$– user64494Jun 30, 2015 at 2:59
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$\begingroup$ Hint: $I=a\ln2+b\ln3+c\ln5$. $\endgroup$– LucianJun 30, 2015 at 3:04
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2$\begingroup$ Mathematica outputs $$28 i \pi + 1536 \log2 - 144 \log3 -2000/5 \log5$$ $\endgroup$– user64494Jun 30, 2015 at 3:07
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$\begingroup$ Have you tried $y_1=x_1+x_2$, $y2_=x_3+x_4$, $y_3=x_1-x_2$, $y_4=x_3-x_4$? $\endgroup$– Michael GaluzaJun 30, 2015 at 3:31
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1$\begingroup$ Is there any reason to believe that it converges? You have non-isolated singularities in the domain (the entirety of the plane $x_1=-x_2, x_3=-x_4$) so at best you can look at PVs. $\endgroup$– Steven StadnickiJun 30, 2015 at 3:56
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1 Answer
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I couldn't resist as if your limits were same for each variable say varying $a$ to $b$, for any $a$ and $b$ , the following trick I saw somewhere, is applicable, but in your case, there is no short cut, I guess.
$$I=\int_{a}^b\int_{a}^b \int_{a}^b \int_{a}^b \frac{x_1-x_2+x_3{-}x_4}{x_1+x_2+x_3+x_4}dx_1dx_2dx_3dx_4 = \int_{a}^b\int_{a}^b \int_{a}^b \int_{a}^b \frac{-x_1-x_2+x_3+x_4}{x_1+x_2+x_3+x_4}dx_1dx_2dx_3dx_4=\int_{a}^b\int_{a}^b \int_{a}^b \int_{a}^b \frac{x_1-x_2-x_3+x_4}{x_1+x_2+x_3+x_4}dx_1dx_2dx_3dx_4 = \int_{a}^b\int_{a}^b \int_{a}^b \int_{a}^b \frac{-x_1+x_2{-}x_3+x_4}{x_1+x_2+x_3+x_4}dx_1dx_2dx_3dx_4=\int_{a}^b\int_{a}^b \int_{a}^b \int_{a}^b \int_{-a}^a \frac{x_1+x_2-x_3-x_4}{x_1+x_2+x_3+x_4}dx_1dx_2dx_3dx_4 = \int_{a}^b\int_{a}^b \int_{a}^b \int_{a}^b \frac{-x_1+x_2+x_3-x_4}{x_1+x_2+x_3+x_4}dx_1dx_2dx_3dx_4$$
Adding them all gives $6I=0 \implies I=0$
P.S- I know it doesn't answer your question, but it was too big/clumsy for a comment.
answered Jun 30, 2015 at 3:24
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$\begingroup$ How about the convergence of the integrals? The above is in Euler's style. $\endgroup$ Jun 30, 2015 at 3:27
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2$\begingroup$ I never worry about convergence until a singularity hits me on the head. $\endgroup$ Jun 30, 2015 at 4:43
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$\begingroup$ @marty cohen: Then all that is built on sand. $\endgroup$ Jun 30, 2015 at 5:26
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1$\begingroup$ Ah, but take that sand, add water, and it turns into mud. Form that mud into blocks, let it dry in the sun, and it turns into bricks which can then be used to construct a magnificent building. $\endgroup$ Jun 30, 2015 at 14:02