Generalization of vector identity $\|\sum_i x_i\|^2 = \sum_i \|x_i\|^2 + \sum_{i \not = j} x_i \cdot x_j$ to integrals?

Question

Let $f:[0,1] \to \mathbb{R}^n$ be smooth. I was wondering if I can reformulate the expression $\|\int_0^1 f(t)dt\|^2$ in a way that mimics the law of cosine vector identity: $\|\sum_i x_i\|^2 = \sum_i \|x_i\|^2 + \sum_{i \not = j} x_i \cdot x_j$. I am aware that $\| \int_0^1 f(t)dt\|^2 \leq \int_0^1 \|f(t)\|^2dt$, however I am searching for an equality rather than an inequality. Does any such identity exist?