How do I change the number of variables in an integral? If I define
$$\phi (k) = \int_{-\infty}^\infty e^{ikx} p(x) dx$$
but I want to define $x=x_1+x_2+x_3+...x_n$, how would I change my variables of integration to do that integral? I ultimately want to show that if
$$\phi_i (k) = \int_{-\infty}^\infty e^{ikx_i} p(x_i) dx$$
then 
$$\phi = \phi_i^n.$$
I know I can appeal to the definition of the expectation value to do this if $p(x)$ is interpreted as a probability distribution, but I want to see this done from the definition of $\phi$ directly, and it's unclear to me how you would change that first integral to be integrated over multiple variables.
Edit: See comment below regarding what I mean that this can be shown by appealing to the definition of expectation values. Sorry if that was unclear.

I think this answer covers most of what you are asking about. What it essentially boils down to is that you have to integrate over $x_1$, $x_2$, . . . and $x_n$. What you haven't mentioned is that your random variables are independent, which means that the joint probability density function factorises into a product:
$$ f(x_1,x_2,\dotsc,x_n) = p(x_1) p(x_2) \dotsm p(x_n). $$
(And if you don't have independence, you can't do anything to split this up.)
Then in your notation you get the simple factorisation
$$ \phi(k) = \idotsint e^{ik(x_1+\dotsb + x_n)} f(x_1,x_2,\dotsc,x_n) dx_1 \dotsm dx_n \\
= \left( \int e^{ikx_1} p(x_1) dx_1 \right) \dotsm \left( \int e^{ikx_1} p(x_n) dx_n \right) = \phi_i^n
$$
