∫_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


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Question Number 206773 by MaruMaru last updated on 24/Apr/24

$\int_{0}^{\infty} \frac{e^{- x^{2}}}{{(x^{2} + \frac{1}{2})}^{2}} d x =$ $I^{2} = \int \int_{D} \frac{e^{- x^{2} - y^{2}}}{{(x^{2} + \frac{1}{2})}^{2} {(y^{2} + \frac{1}{2})}^{2}} d A$ $x = r \cos (θ) y = r \sin (θ)$ $J =∣ \frac{\partial (x, y)}{\partial (r, θ)} ∣ d r d θ = r d r d θ$ $\int \int_{D} \frac{{r e}^{- r^{2}}}{{(r^{2} \cos^{2} (θ) + \frac{1}{2})}^{2} {(r^{2} \sin^{2} (θ) + \frac{1}{2})}^{2}} d r d θ$
Commented by MaruMaru last updated on 24/Apr/24

$n e x t . . . ? ? ?$
Commented by Frix last updated on 24/Apr/24

$Differnt paths see question 206549$
Commented by MaruMaru last updated on 25/Apr/24

$w o w t h x$
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