I'm currently studying signal and system, and the homework requires to find the impulse response of the system $$y(t)=\int_{-\infty}^t(t-\tau+2)x(\tau)d\tau$$
I want to plug in $x(\tau)=\delta(\tau)$: $$y(t)=\int_{-\infty}^t(t-\tau+2)\delta(\tau)d\tau$$ and I have no idea how to deal with it. I want to use some property mentioned in the textbook (like the sifting property) but now the variables are just messed with each other and I'm confused.
Can anybody help me?
In general, for any function $g(x)$ (continous at $x=0$) we have $\int_{-\infty}^\infty g(x)\delta(x)dx = g(0)$
Then, define $$g(\tau) = \begin{cases} 0 & \text{if } \tau > t \\ t-\tau+2 & \text{if } \tau \le t \end{cases} $$
so we can write
$$y(t)=\int_{-\infty}^t(t-\tau+2)\delta(\tau)d\tau =\int_{-\infty}^\infty g(\tau) \delta(\tau)d\tau = g(0) = \begin{cases} 0 & \text{if } t <0 \\ t +2 & \text{if } t > 0 \end{cases} $$