Density of $X_1 + \cdots+X_n$ when $X_i$'s are independent $U(-1,1)$ variables How can we show for independent random variables uniformly distributed over $(1,-1)$ that $X_1 + \cdots+X_n$
has density 
$$\pi^{-1}\int^\infty_0 \left(\frac{\sin t}{t}\right)^n \cos tx \; dt \textrm{ for }n \geq 2\text{?} $$
This is   problem 26.6 from Billingsley  "Chapter : CONVERGENCE OF DISTRIBUTIONS" (3rd edition).
I would really appreciate it, if you show it analytically .

Hint
If $X,Y$ are independent and continuous, then $X+Y$ has density function $$f_X*f_Y(t)=\int_{\mathbb R}f_X(t-u)f_Y(u)\,\mathrm d u.$$
