Request
I am very new to this so please bear with me. I cannot duplicate the answer in the book. I believe I may be making a methodical error. Please correct it for me.
Given:
Find the convolution of $f(t)=1$ and $g(t)=1$.
$$h(t)=(f*g)(t)=\int_0^t f(\tau)g(t-\tau)d\tau$$
My Solution:
$$h(t)=(1*1)(t)=\int_0^t 1\cdot (1-\tau)d\tau=\tau-\frac{\tau^2}{2}|_0^t=t^-\frac{t^2}{2} = \frac{t}{2}$$
Answer in Text:
$$h(t)=t$$
Best Answer
The problem is that you're using $1-\tau$ as part of the integrand. However, if $g(t)=1$ then $g(t-\tau)$ also equals $1$, and the integral is simply
$$\int_0^tf(\tau)g(t-\tau)d\tau=\int_0^t1\cdot 1\,d\tau=\int_0^td\tau=t$$