LTI system's response to input's integral

Question

Suppose that $H$ is a linear time invariant system. If the response of $H$ to the input $u$ is $y$, then what is its response to integral of $u$?

Answer 1

Since $H$ is an LTI system, it has an impulse-response, say $h(t)$ and for $u(t)$ as the input signal: $$y(t)=h(t)*u(t)$$ where $*$ represents the convolution. Note that the above equation is the inherent property of LTI systems. Now since the integral and convolution are both linear operators, if $u_I=\int u$ then $$\begin{align}y_I(t)&=h(t)*u_I(t)\\ &=h(t)*\int_{t_0}^t u(\tau)d\tau\\ &=\int_{t_0}^t h(\tau)*u(\tau)d\tau=\int_{t_0}^t y(\tau)d\tau \end{align}$$ Or in other words $y_I=\int y$