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.
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$\begingroup$ that is... basic math, not signal processing. That's the integration variable; I'm sure you've seen an integral before? $\endgroup$– Marcus MüllerNov 10 '20 at 19:58
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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
$\int_0^tm(t)dt\qquad(???)\tag{1}$
doesn't make sense, whereas
$$\int_0^tm(x)dx\tag{2}$$
does.
Take a look at this question and its answers over at math.SE.