Why isn't this a right expression for FM?
An FM-modulated signal looks like
$$A_c \cos\left(\varphi\left(t\right)\right)$$

At time \$t\$, its frequency is
$${1 \over 2 \pi}{\mathrm{d}\varphi \over \mathrm{d}t}$$

And you want it to be
$$f_c + K m(t)$$

So you must have
$$\varphi(t) = \varphi_0 + \int_0^t f_c + K m(u) \mathrm{d}u
             = \varphi_0 + f_c t + K \int_0^t m(u) \mathrm{d}u$$

Hence, the expression for an FM-modulated signal is
$$A_c \cos\left(\varphi_0 + f_c t + K \int_0^t m(u) \mathrm{d}u\right)$$