series and definite integral

    \begin{align*} \ln(2)=1-\fr{2}+\fr{3}-\fr{4}+\fr{5}-\dots&=\int\limits_0^1 \frac{2x}{1+x^2} \;\text{d}x = 0.693147\dots \\ \frac{\pi}{4}=1-\fr{3}+\fr{5}-\fr{7}+\fr{9}-\dots&=\int\limits_0^1 \fr{1+x^2} \;\text{d}x = 0.785398\dots \\ \intertext{Catalan's constant} G=\fr{1^2}-\fr{3^2}+\fr{5^2}-\fr{7^2}+\fr{9^2}-\dots&=\int\limits_1^\infty \frac{\ln(x)}{1+x^2} \;\text{d}x=0.9159655\dots \end{align*}


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