# What does the variable $\tau$ mean in frequency modulation?

Frequency modulation formula:

$$x_{FM}(t) = A_{c}\cos\left(2\pi f_{c} t+ 2\pi k_{f}\displaystyle\int_0^{t}m(\tau)d\tau+\phi_{0}\right)$$

Its change from $$m(t)$$ to $$m(\tau)$$.
What is $$\tau$$? I wonder what it means.

• that is... basic math, not signal processing. That's the integration variable; I'm sure you've seen an integral before? Nov 10 '20 at 19:58

It's just a dummy integration variable. The time variable $$t$$ is the upper integration limit, so you can't use it as the integration variable. An expression like
$$\int_0^tm(t)dt\qquad(???)\tag{1}$$
$$\int_0^tm(x)dx\tag{2}$$