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0-e-x-2-x-2-1-2-2-dx-I-2-D-e-x-2-y-2-x-2-1-2-2-y-2-1-2-2-dA-x-rcos-y-rsin-J-x-y-r-drd-rdrd-D-re-r-2-r-2-cos-2-




Question Number 206773 by MaruMaru last updated on 24/Apr/24
∫_0 ^∞  (e^(−x^2 ) /((x^2 +(1/2))^2 ))dx=  I^2 =∫∫_( D)  (e^(−x^2 −y^2 ) /((x^2 +(1/2))^2 (y^2 +(1/2))^2 ))dA  x=rcos(θ)  y=rsin(θ)  J=∣((∂(x,y))/(∂(r,θ)))∣drdθ=rdrdθ  ∫∫_( D)  ((re^(−r^2 ) )/((r^2 cos^2 (θ)+(1/2))^2 (r^2 sin^2 (θ)+(1/2))^2 ))drdθ
$$\int_{\mathrm{0}} ^{\infty} \:\frac{{e}^{−{x}^{\mathrm{2}} } }{\left({x}^{\mathrm{2}} +\frac{\mathrm{1}}{\mathrm{2}}\right)^{\mathrm{2}} }{dx}= \\ $$$${I}^{\mathrm{2}} =\int\int_{\:\boldsymbol{\mathcal{D}}} \:\frac{{e}^{−{x}^{\mathrm{2}} −{y}^{\mathrm{2}} } }{\left({x}^{\mathrm{2}} +\frac{\mathrm{1}}{\mathrm{2}}\right)^{\mathrm{2}} \left({y}^{\mathrm{2}} +\frac{\mathrm{1}}{\mathrm{2}}\right)^{\mathrm{2}} }{dA} \\ $$$${x}={r}\mathrm{cos}\left(\theta\right)\:\:{y}={r}\mathrm{sin}\left(\theta\right) \\ $$$${J}=\mid\frac{\partial\left({x},{y}\right)}{\partial\left({r},\theta\right)}\mid{drd}\theta={rdrd}\theta \\ $$$$\int\int_{\:\boldsymbol{\mathcal{D}}} \:\frac{{re}^{−{r}^{\mathrm{2}} } }{\left({r}^{\mathrm{2}} \mathrm{cos}^{\mathrm{2}} \left(\theta\right)+\frac{\mathrm{1}}{\mathrm{2}}\right)^{\mathrm{2}} \left({r}^{\mathrm{2}} \mathrm{sin}^{\mathrm{2}} \left(\theta\right)+\frac{\mathrm{1}}{\mathrm{2}}\right)^{\mathrm{2}} }{drd}\theta \\ $$
Commented by MaruMaru last updated on 24/Apr/24
next...???
$${next}…??? \\ $$
Commented by Frix last updated on 24/Apr/24
Differnt paths see question 206549
$$\mathrm{Differnt}\:\mathrm{paths}\:\mathrm{see}\:\mathrm{question}\:\mathrm{206549} \\ $$
Commented by MaruMaru last updated on 25/Apr/24
wow thx
$${wow}\:{thx} \\ $$

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