The integral you provided is:

$\int_0^1 1 \, dx$

This represents the area under the constant function $f(x) = 1$ between $x = 0$ and $x = 1$.

Solution:

$\int_0^1 1 \, dx = [x]_0^1 = 1 - 0 = 1$

Thus, the value of the integral is $\boxed{1}$.

Would you like further details or clarification?

Here are some related questions:

1. How would you calculate the integral of a constant function over different intervals?
2. What is the general rule for integrating constant functions?
3. How does the area under a curve relate to integrals?
4. What happens if the constant function is negative, say $f(x) = -1$?
5. How do you approach integrals of piecewise constant functions?

Tip: The integral of a constant function $f(x) = c$ over an interval $[a, b]$ is simply $c(b - a)$.