A cute integral with golden ratio

 Show that

$$\int_{0}^{\infty} \frac{dx}{(1 + x^{\varphi})^{\varphi}} = 1$$

where $\varphi = \frac{\sqrt{5} + 1}{2}$ is the golden ratio.

Scroll down for solution:

Using the substitution $x^{-\varphi} = t$ and the identity $\varphi^2 = \varphi + 1$ we get that

$$dx = -\frac{1}{\varphi} t^{-\varphi} dt$$

Thus the integral becomes

$$\frac{1}{\varphi} \int_{0}^{\infty} \frac{dt}{(1 + t)^{\varphi}}$$

Which is easy to calculate.