We need to evaluate the definite integral of $\frac{1}{1+x^{60}}$ from $0$ to $\infty$. That is, we want to find the value of $\int_{0}^{\infty} \frac{1}{1+x^{60}} dx$.

1. Problem Description

We need to evaluate the definite integral of $\frac{1}{1+x^{60}}$ from $0$ to $\infty$. That is, we want to find the value of $\int_{0}^{\infty} \frac{1}{1+x^{60}} dx$.

2. Solution Steps

We can use the following general formula for integrals of the form $\int_0^\infty \frac{x^{m-1}}{1+x^n} dx$:

$\int_0^\infty \frac{x^{m-1}}{1+x^n} dx = \frac{\pi}{n \sin(\frac{m\pi}{n})}$

In our case, we have $m-1 = 0$, so $m=1$. And $n=60$. Plugging these values into the formula, we have:

$\int_{0}^{\infty} \frac{1}{1+x^{60}} dx = \int_{0}^{\infty} \frac{x^{1-1}}{1+x^{60}} dx = \frac{\pi}{60 \sin(\frac{1\pi}{60})} = \frac{\pi}{60 \sin(\frac{\pi}{60})}$

Therefore, the integral evaluates to $\frac{\pi}{60 \sin(\frac{\pi}{60})}$.

3. Final Answer

$\frac{\pi}{60 \sin(\frac{\pi}{60})}$