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Evaluate $\int_{0}^{1}\int_{0}^{\sqrt{1+x^2}}\frac{dy\,dx}{(1+x^2+y^2)}$ .
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$ \int_{x=0}^{1} \int_{y=0}^{\sqrt{1+x^2}} \frac{dy dx} {1+ x^2 + y^2 } \\ $

$ = \int_{x=0}^{1} \int_{y=0}^{\sqrt{1+x^2}} \frac{dy} {(1+ x^2 )+ y^2 } dx \\ $

$ = \int_{x=0}^{1} \frac{1}{\sqrt{1+x^2}}\left [tan^{-1} \frac{y} {\sqrt{1+ x^2} } \right]_0^{\sqrt{1 + x^2}} dx \\ $

$ = \int_{x=0}^{1} \frac{1}{\sqrt{1+x^2}} \left [ \frac{\pi} {4 } \right]dx \\ $

$ = \frac{\pi}{4} \left [log (x + \sqrt{1+ x^2} \right]_0^1 \\ $

$ = \frac{\pi}{4} \left [log (1 + \sqrt{2} \right] \\ $

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