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Question Number 82872 by abdomathmax last updated on 25/Feb/20

1) f i n d \int \int_{W} \frac{x d x}{a^{2} + x^{2} + y^{2}} w i t h

W_{a} \to x^{2} + y^{2} ⩽ a^{2} a n d x > 0 (a > 0)

2) c a l c u l a t e \int \int_{W_{1}} \frac{x d x}{x^{2} + y^{2} + 1}

Commented by mathmax by abdo last updated on 25/Feb/20

1) w e u s e t h e d i f f e o m o r p h i s m (r, θ) \to (x, y) / x = r c o s θ

a n d y = r s i n θ w e h a v e x^{2} + y^{2} ⩽ a^{2} \Rightarrow 0 ⩽ r ⩽ a x > 0 \Rightarrow θ \in] - \frac{π}{2}, \frac{π}{2} [

\Rightarrow \int \int_{W_{a}} \frac{x d x d y}{a^{2} + r^{2}} = \int_{0}^{a} \int_{- \frac{π}{2}}^{\frac{π}{2}} \frac{r c o s θ r d r d θ}{a^{2} + r^{2}}

\int_{0}^{a} \frac{r^{2}}{r^{2} + a^{2}} d r \times \int_{- \frac{π}{2}}^{\frac{π}{2}} c o s θ d θ = 2 \int_{0}^{a} \frac{r^{2}}{r^{2} + a^{2}} d r

= 2 \int_{0}^{a} \frac{r^{2} + a^{2} - a^{2}}{r^{2} + a^{2}} d r = 2 a - 2 a^{2} \int_{0}^{a} \frac{d r}{r^{2} + a^{2}} (c h . r = a x)

= 2 a - 2 a^{2} \int_{0}^{1} \frac{a d x}{a^{2} (1 + x^{2})} = 2 a - 2 a {[a r c t a n x]}_{0}^{1}

= 2 a - 2 a \times \frac{π}{4} = (2 - \frac{π}{2}) a

2) \int \int_{W_{1}} \frac{x d x}{x^{2} + y^{2} + 1} =_{a = 1} 2 - \frac{π}{2}