Let $p$ be a probability on $\mathbb{R}$ and 
$$f(t):=\int\_\mathbb{R}\cos(tx)dp(x).$$
I want to show that if $f''(0)$ exists then $$f''(0)=\lim\_{t\to 0}2\frac{f(t)-1}{t^2} \: \:(\star).$$

By differentiation under the integral sign, I obtain
$$f''(t)=\int\_\mathbb{R}-x^2\cos(tx)dp(x).$$
Now, it seems that I need a sort of "integration by parts for probabilities", but I don't know if something similar exists.

So what's the fastest way to deduce $(\star)$?