PAIR OF ARCTAN INTEGRALS

In this post we will evaluate the follwoing pair of integrals. Click here to the proof:



\begin{align*} \int_0^\infty \frac{\arctan(x)}{\sqrt{x}(1+x^2)}\,dx=\frac{\pi^2}{4\sqrt{2}}-\frac{\pi \ln(2)}{2\sqrt{2}} \end{align*}


\begin{align*} \int_0^\infty \frac{\sqrt{x}\arctan(x)}{(1+x^2)}\,dx=\frac{\pi^2}{4\sqrt{2}}+\frac{\pi \ln(2)}{2\sqrt{2}} \end{align*}

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