let a>0 calculate ∫∫_(x^2 +y^2 ≤3) (1/(2 +x^2 +y^2 ))dxdy.


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Question Number 34713 by abdo mathsup 649 cc last updated on 10/May/18

$l e t a > 0 c a l c u l a t e \int \int_{x^{2} + y^{2} ⩽ 3} \frac{1}{2 + x^{2} + y^{2}} d x d y .$
Commented bymath khazana by abdo last updated on 11/May/18
$let ude?the diffeomorphisme x=r cosθ , y = r sinθ with 0≤r≤(√3) and −π≤θ≤π I =∫∫_(0≤r≤(√(3 )) and −π≤ θ≤π) (1/(2 +r^2 )) rdr dθ = ∫_0 ^(√3) (r/(2+r^2 )) dr. ∫_(−π) ^π dθ = 2π ∫_0 ^(√3) (r/(2+r^2 ))dr =π[ln(2+r^2 )]_0 ^(√3) = π{ln(5) −ln(2)} .$
$l e t u d e ? t h e d i f f e o m o r p h i s m e x = r c o s θ,$ $y = r s i n θ w i t h 0 ⩽ r ⩽ \sqrt{3} a n d - π ⩽ θ ⩽ π$ $I = \int \int_{0 ⩽ r ⩽ \sqrt{3} a n d - π ⩽ θ ⩽ π} \frac{1}{2 + r^{2}} r d r d θ$ $= \int_{0}^{\sqrt{3}} \frac{r}{2 + r^{2}} d r . \int_{- π}^{π} d θ = 2 π \int_{0}^{\sqrt{3}} \frac{r}{2 + r^{2}} d r$ $= π {[l n (2 + r^{2})]}_{0}^{\sqrt{3}} = π {l n (5) - l n (2)} .$
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