Solving $\int_0^\infty\dfrac{dx}{(1+x^n)^n}=1$

Question

A friend has challenged me to find the value of $n$ such that $\displaystyle\int_0^\infty\dfrac{dx}{(1+x^n)^n}=1.$ I know that the value of $n$ is equal to $\phi,$ but I do not know how to prove this.

Answer 1

HINT:

Enforce the substitution $x=t^{1/n}$. Then, verify a resulting equation.

SPOILER ALERT: Scroll over the highlighted area to reveal the solution

Let $I(n)$ be the integral given by $$I(n)=\int_0^\infty \frac{1}{(1+x^n)^n}\,dx \tag 1$$First note that the integral converges for $n>1$ and diverges otherwise. Next, enforcing the substitution $x= t^{1/n}$ in $(1)$ reveals $$\begin{align}I(n)&=\frac1n \int_0^\infty \frac{t^{1/n-1}}{(1+t)^n}\,dt\\\\&=\frac1n B(1/n,n-1/n)\\\\&=\frac{\Gamma(1/n)\Gamma(n-1/n)}{n\Gamma(n)}\end{align}$$Therefore, if $I(n)=1$, then $$\frac1n \Gamma(1/n)\Gamma(n-1/n)=\Gamma(n) \tag 2$$If $n=\phi>1$ in $(2)$, then $n-1/n=1$ and $(2)$ states that $\Gamma(\phi)=\frac1\phi \Gamma(1/\phi)=\Gamma(1+1/\phi)$. Since $1+1/\phi=\phi$, we are done!

Answer 2

I don't see how do we do that unless we bring the whole thing to Beta function with some clever (or maybe trivial) substitution.

Say, $t={1\over1+x^n}$. Then $x=\sqrt[n]{{1\over t}-1}$. Then $dx=-{dt\over n t^2({1/t}-1)^{1-1/n}}$, and $$\int_0^\infty\dfrac{dx}{(1+x^n)^n}=-\int_1^0t^n\left({1\over t}-1\right)^{{1\over n}-1}{dt\over n\cdot t^2}={1\over n}\int_0^1t^{n-1-{1\over n}}(1-t)^{{1\over n}-1}dt={1\over n}B\left(n-{1\over n},{1\over n}\right)={1\over n}\cdot{\Gamma\left(n-{1\over n}\right)\Gamma\left({1\over n}\right)\over \Gamma(n)}={\Gamma\left(n-{1\over n}\right)\Gamma\left(1+{1\over n}\right)\over \Gamma(n)}$$

At this point, it is pretty easy to verify directly that $n=\varphi$ fits. (Note that $\varphi-{1\over\varphi}=1$ and $1+{1\over\varphi}=\varphi$).

What could we do without that afterknowledge, I wonder...

Answer 3

The easy part. One may observe that, by the change of variable $x \to \dfrac1x$, one gets $$ \begin{align} \int_0^\infty\frac{dx}{(1+x^\phi)^\phi}&=\int_0^\infty\frac{x^{\phi^2-2}}{(1+x^\phi)^\phi}\:dx \\\\&=\int_0^\infty\frac{x^{\phi-1}}{(1+x^\phi)^\phi}\:dx \\\\&=\frac1{\phi}\int_0^\infty\frac{(1+x^{\phi})'}{(1+x^\phi)^\phi}\:dx \\\\&=\frac1{\phi}\int_1^\infty u^{-\phi}\:du \\\\&=\frac{1}{\phi(\phi-1)} \\\\&=1 \end{align} $$ if $\phi>1$ and $\phi^2=\phi+1$.