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Question Number 209624 by mnjuly1970 last updated on 16/Jul/24

I = \int_{0}^{\infty} \int_{0}^{\infty} \frac{1}{1 + x^{2} + y^{2} + x^{2} y^{2}} d x d y = ?

u s i n g p o l a r s y s t e m \dots

Answered by mr W last updated on 16/Jul/24

= \int_{0}^{\infty} \int_{0}^{\infty} \frac{1}{(1 + x^{2}) (1 + y^{2})} d x d y

= {[\int_{0}^{\infty} \frac{1}{1 + x^{2}} d x]}^{2}

= {({[\tan^{- 1} x]}_{0}^{\infty})}^{2}

= {(\frac{π}{2})}^{2} = \frac{π^{2}}{4}

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