This question seems basic but I could not find an answer. I have seen the inequality $\left\|\int_a^b x(t) dt \right\| \leq \int_a^b \left\| x(t) \right\| dt $ where $x(t) \in \mathbb{R}^n$ is a vector function and $\|\cdot\|$ is a vector norm, and $a < b$.
I wonder if this also holds for matrices with induced norm, that is $\left\|\int_a^b X(t) dt \right\| \leq \int_a^b \left\| X(t) \right\| dt $ where $X(t)$ is a matrix function and $\|\cdot\|$ is an induced matrix norm, and $a < b$. If it is true, is there any reliable citation source?