How can one show that $$ f * g = t^{m+n+1} \int_0^1 u^m(1-u)^n du $$ where $f(t) = t^m$, $g(t) = t^n$ and $$f*g = \int_0^t f(\tau)g(t-\tau)d\tau = \int_0^t f(t-\tau)g(\tau)d\tau $$
I tried with the binomial theorem on $(t-\tau)^n$ but to no avail. Unable to progress from here.
$$ f \star g(t) = \int_0^t \tau^m (t-\tau)^n\, d\tau $$
Now with $\tau =ut$, you get $$ f \star g(t) = \int_0^1 (ut)^m (t-ut)^n t\,du = t^{m+n+1}\int_0^1 u^m (1-u)^n \,du $$