Variation of Ahmed's integral
$$\int_{0}^{1}{\mathrm dx\over 1+x^2}\cdot{\tan^{-1}\left(\sqrt{x^2+2}\cdot{x+1\over x^2-x+2}\right)\over \sqrt{x^2+2}}={13\pi^2\over 288}\tag1$$
Making an attempt:
I can't think of what to do. It is too complicate.
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asked Apr 14, 2017 at 6:09
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$\begingroup$ Which result is correct: $13\pi^2$ and $5\pi^2$? $\endgroup$– Hoc NgoApr 14, 2017 at 6:29
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$\begingroup$ They are the same just different way of writing it $\endgroup$– gymbvghjkgkjkhgfklApr 14, 2017 at 6:49
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1$\begingroup$ Where does this integral come from? $\endgroup$– FDPApr 14, 2017 at 9:35
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$\begingroup$ Variation of Ahmed integral. $\endgroup$– gymbvghjkgkjkhgfklApr 14, 2017 at 9:40
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1$\begingroup$ Where do the fraction $\dfrac{x+1}{x^2-x+2}$ come from? I mean if i replace it by, at random, $\dfrac{x+1}{x(x+2)}$ no such interresting result is found. Godess Namagiri gives you that integral in your dreams? :) $\endgroup$– FDPApr 14, 2017 at 10:08
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1 Answer
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One may observe that, for $x \in [0,1]$, $$ \arctan\left(\sqrt{x^2+2}\cdot{x+1\over x^2-x+2}\right)=\arctan\left({1\over \sqrt{x^2+2}}\right)+\arctan\left({x\over \sqrt{x^2+2}}\right) $$ then $$ I:=\int_0^1 \frac{1}{(1+x^2)\sqrt{x^2+2}}\arctan\left(\sqrt{x^2+2}\cdot{x+1\over x^2-x+2}\right)dx=I_1+I_2 $$with $$ \begin{align} I_1:&=\int_0^1 \frac{1}{(1+x^2)\sqrt{x^2+2}}\arctan\left({1\over \sqrt{x^2+2}}\right)dx \\&=\int_0^1 \frac{1}{(1+x^2)\sqrt{x^2+2}}\left(\frac \pi2-\arctan\sqrt{x^2+2}\right)dx \\&=\frac \pi2\int_0^1 \frac{1}{(1+x^2)\sqrt{x^2+2}}-\int_0^1 \frac{\arctan\sqrt{x^2+2}}{(1+x^2)\sqrt{x^2+2}}\:dx \\&=\frac{\pi^2}{12}-\frac{5\pi^2}{96} \\&=\frac{\pi^2}{32} \end{align} $$ where the latter integral is Ahmed's evaluation.
On the other hand we have $$ \begin{align} I_2:&=\int_0^1 \frac{1}{(1+x^2)\sqrt{x^2+2}}\cdot\arctan\left({x\over \sqrt{x^2+2}}\right)dx \\&=\int_1^\infty \frac{u}{(1+u^2)\sqrt{1+2u^2}}\cdot\arctan\left({1\over \sqrt{1+2u^2}}\right)du \quad (u=1/x) \\&=\frac12\int_1^\infty \frac{1}{(1+v)\sqrt{1+2v}}\cdot\arctan\left({1\over \sqrt{1+2v}}\right)dv \qquad (v=u^2) \\&=-\int_1^\infty \left[\arctan\left({1\over \sqrt{1+2v}}\right)\right]'\arctan\left({1\over \sqrt{1+2v}}\right)dv \\&=-\left[\frac12\cdot\arctan^2\left({1\over \sqrt{1+2v}}\right)\right]_1^\infty \\&=\frac{\pi^2}{72}. \end{align} $$ Finally,
$$ \int_0^1 \frac{1}{(1+x^2)\sqrt{x^2+2}}\arctan\left(\sqrt{x^2+2}\cdot{x+1\over x^2-x+2}\right)dx=\frac{\pi^2}{32}+\frac{\pi^2}{72}=\color{blue}{\frac{13\pi^2}{288}} $$
as announced.
answered Apr 14, 2017 at 13:25
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$\begingroup$ Nice answer. I understand now why Latte wasn't answering my question about the source of this integral, magicians don't reveal their tricks. $\endgroup$– FDPApr 14, 2017 at 15:17
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3$\begingroup$ $\displaystyle \int \frac{\arctan\left(\frac{x}{\sqrt{x^2+2}}\right)}{(1+x^2)\sqrt{x^2+2}}dx=\dfrac{1}{2}\arctan^2\left(\frac{x}{\sqrt{x^2+2}}\right)+\text{constant}$ $\endgroup$– FDPApr 14, 2017 at 15:41