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Question Number 220896 by Nicholas666 last updated on 20/May/25

∫∫∫_( [0,∞]^( 3) ) ((x^2 y^2 z^2 )/((1 + x^2 + y^2 + z^2 )^5 )) dxdydz

$$ \\ $$$$\:\:\:\int\int\int_{\:\left[\mathrm{0},\infty\right]^{\:\mathrm{3}} } \frac{{x}^{\mathrm{2}} {y}^{\mathrm{2}} {z}^{\mathrm{2}} }{\left(\mathrm{1}\:+\:{x}^{\mathrm{2}} \:+\:{y}^{\mathrm{2}} \:+\:{z}^{\mathrm{2}} \right)^{\mathrm{5}} }\:{dxdydz}\:\:\:\:\:\:\:\:\: \\ $$$$ \\ $$

Answered by deleted50 last updated on 21/May/25

∫∫∫_R^3 (((xyz)^2 )/((1+x^2 +y^2 +z^2 )^5 )) dxdydz= x=rsin(θ)cos(ρ) y=rsin(θ)sin(ρ) z=rcos(θ) ∣∣J∣∣=∣∣(∂x/∂r)∣∣ dr → r^2 sin(θ)drdθdρ ∫∫∫_R^3 ((r^8 sin^5 (θ)cos^2 (θ)sin^2 (ρ)cos^2 (ρ))/((r^2 +1)^5 )) drdθdρ =((35)/(256))π∫∫ sin^5 (θ)cos^2 (θ)sin^2 (ρ)cos^2 (ρ) dθdρ ((35)/(256))π∙((16)/(105)) ∫ sin^2 (ρ)cos^2 (ρ)dρ=(π^2 /(1536))

$$\int\int\int_{\mathbb{R}^{\mathrm{3}} } \:\:\frac{\left({xyz}\right)^{\mathrm{2}} }{\left(\mathrm{1}+{x}^{\mathrm{2}} +{y}^{\mathrm{2}} +{z}^{\mathrm{2}} \right)^{\mathrm{5}} }\:\mathrm{d}{x}\mathrm{d}{y}\mathrm{d}{z}= \\ $$$${x}={r}\mathrm{sin}\left(\theta\right)\mathrm{cos}\left(\rho\right) \\ $$$${y}={r}\mathrm{sin}\left(\theta\right)\mathrm{sin}\left(\rho\right) \\ $$$${z}={r}\mathrm{cos}\left(\theta\right) \\ $$$$\mid\mid\boldsymbol{{J}}\mid\mid=\mid\mid\frac{\partial\boldsymbol{\mathrm{x}}}{\partial\boldsymbol{\mathrm{r}}}\mid\mid\:\mathrm{d}\boldsymbol{\mathrm{r}}\:\rightarrow\:{r}^{\mathrm{2}} \mathrm{sin}\left(\theta\right)\mathrm{d}{r}\mathrm{d}\theta\mathrm{d}\rho \\ $$$$\int\int\int_{\mathcal{R}^{\mathrm{3}} } \:\frac{{r}^{\mathrm{8}} \mathrm{sin}^{\mathrm{5}} \left(\theta\right)\mathrm{cos}^{\mathrm{2}} \left(\theta\right)\mathrm{sin}^{\mathrm{2}} \left(\rho\right)\mathrm{cos}^{\mathrm{2}} \left(\rho\right)}{\left({r}^{\mathrm{2}} +\mathrm{1}\right)^{\mathrm{5}} }\:\mathrm{d}{r}\mathrm{d}\theta\mathrm{d}\rho \\ $$$$=\frac{\mathrm{35}}{\mathrm{256}}\pi\int\int\:\:\mathrm{sin}^{\mathrm{5}} \left(\theta\right)\mathrm{cos}^{\mathrm{2}} \left(\theta\right)\mathrm{sin}^{\mathrm{2}} \left(\rho\right)\mathrm{cos}^{\mathrm{2}} \left(\rho\right)\:\mathrm{d}\theta\mathrm{d}\rho \\ $$$$\frac{\mathrm{35}}{\mathrm{256}}\pi\centerdot\frac{\mathrm{16}}{\mathrm{105}}\:\int\:\:\mathrm{sin}^{\mathrm{2}} \left(\rho\right)\mathrm{cos}^{\mathrm{2}} \left(\rho\right)\mathrm{d}\rho=\frac{\pi^{\mathrm{2}} }{\mathrm{1536}} \\ $$

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