Question Number 78701 by abdomathmax last updated on 20/Jan/20
$${calculate}\:\int\int_{{D}} \:\:\frac{{dxdy}}{\left(\mathrm{1}+{x}^{\mathrm{2}} \right)\left(\mathrm{1}+{y}^{\mathrm{2}} \right)} \\ $$$${with}\:{D}\:=\left\{\left({x},{h}\right)\in{R}^{\mathrm{2}} \:\:/\mathrm{0}\leqslant{x}\leqslant\mathrm{1}\:{and}\:\:\mathrm{0}\leqslant{y}\:\leqslant{x}\right\} \\ $$
Commented by john santu last updated on 20/Jan/20
$$=\underset{\mathrm{0}} {\overset{{x}} {\int}}\:\underset{\mathrm{0}} {\overset{\mathrm{1}} {\int}}\:\left(\frac{{dx}}{\left(\mathrm{1}+{x}^{\mathrm{2}} \right)\left(\mathrm{1}+{y}^{\mathrm{2}} \right)}\right){dy} \\ $$$$=\:\underset{\mathrm{0}} {\overset{{x}} {\int}}\:\frac{\mathrm{1}}{\left(\mathrm{1}+{y}^{\mathrm{2}} \right)}\:.\left(\mathrm{tan}^{−\mathrm{1}} \left({x}\right)\right)\mid_{\mathrm{0}} ^{\mathrm{1}} \:\:{dy} \\ $$$$=\:\frac{\pi}{\mathrm{4}}\int_{\mathrm{0}} ^{{x}} \:\frac{\mathrm{1}}{\left(\mathrm{1}+{y}^{\mathrm{2}} \right)}\:{dy}\: \\ $$$$=\:\frac{\pi}{\mathrm{4}}\:\left\{\mathrm{tan}^{−\mathrm{1}} \left({y}\right)\:\mid_{\mathrm{0}} ^{{x}} \:\right\}\:=\frac{\pi}{\mathrm{4}}.\mathrm{tan}^{−\mathrm{1}} \left({x}\right) \\ $$