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Is it possible to evaluate the following integral in a closed form? $$\int_0^\infty\frac{\sqrt[\phi]{x}\ \arctan x}{\left(x^\phi+1\right)^2}dx,$$ where $\phi$ is the golden ratio: $$\phi=\frac{1+\sqrt{5}}{2}.$$

Laila Podlesny
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2 Answers2

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Since $\frac1\phi=\phi-1$, we get $$ \begin{align} \int_0^\infty\frac{\sqrt[\phi]{x}\,\arctan(x)}{\left(x^\phi+1\right)^2}\mathrm{d}x &=\int_0^\infty\frac{x^\phi\arctan(x)}{\left(x^\phi+1\right)^2}\frac{\mathrm{d}x}{x}\tag{1}\\ &=\int_0^\infty\frac{x^\phi(\frac\pi2-\arctan(x))}{\left(x^\phi+1\right)^2}\frac{\mathrm{d}x}{x}\tag{2} \end{align} $$ Average $(1)$ and $(2)$ to get $$ \begin{align} \int_0^\infty\frac{\sqrt[\phi]{x}\,\arctan(x)}{\left(x^\phi+1\right)^2}\mathrm{d}x &=\frac\pi4\int_0^\infty\frac{x^\phi}{\left(x^\phi+1\right)^2}\frac{\mathrm{d}x}{x}\tag{3}\\ &=\frac\pi{4\phi}\int_0^\infty\frac{x}{\left(x+1\right)^2}\frac{\mathrm{d}x}{x}\tag{4}\\ &=\frac\pi{4\phi}\tag{5} \end{align} $$ Explanation:

$(1)$: $\frac1\phi=\phi-1$
$(2)$: Substitute $x\mapsto\frac1x$
$(3)$: Average $(1)$ and $(2)$
$(4)$: Substitute $x\mapsto x^{1/\phi}$
$(5)$: $\int_0^\infty\frac{\mathrm{d}x}{(x+1)^2}=\left[-\frac1{x+1}\right]_0^\infty=1$

robjohn
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    wow you're really fast . (+1) – what'sup Oct 09 '13 at 19:38
  • Won't the limits be reversed? Or if remaining the same, there should be a minus sign. Did the inside simplification of the integral require a minus sign? (I'm not able to reach step 2 from step 1 after suggested substitution; could you please elaborate?) – Panglossian Oporopolist Dec 28 '16 at 09:02
  • $$ \begin{align} \int_0^\infty f(x)\,\frac{\mathrm{d}x}{x} &= \int_\infty^0 f\left(\frac1x\right)\left(-\frac{\mathrm{d}x}{x}\right)\\ &=\int_0^\infty f\left(\frac1x\right)\frac{\mathrm{d}x}{x} \end{align} $$ – robjohn Dec 28 '16 at 10:48
  • Sigh.. Can't believe I forgot that.. thank you :) – Panglossian Oporopolist Dec 31 '16 at 08:56
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robjohn's result can be generalized to all real $a\ne0$: $$\int_0^\infty\frac{x^{a-1}\arctan x}{(x^a+1)^2}dx=\frac\pi{4\,|a|}.$$

Vladimir Reshetnikov
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    The original question would translate as $$ \int_0^\infty\frac{x^{1/a}\arctan(x)}{(x^a+1)^2}\mathrm{d}x $$ It is $a=\phi$ that gives $1/a=a-1$. However, if you start from my second step, then your statement is true. – robjohn Oct 10 '13 at 13:01