I'm having quite a hard time trying to simplify the following equations:
$sinc(t)\delta(t) \\ u(t)u(t) $
(by u(t) I mean the unit step function)
also, the integral $ \int^{+\infty}_{-\infty} cos(t)\delta(t) \, dt = 1 $ raises some questions
I'm hoping you guys can help me out!
The Dirac Delta is only implicitely defined with these equations $$f(t)\cdot \delta(t) = f(0)\cdot\delta(t)$$ and $$\int_{-\infty}^{\infty}\delta(t)\mathrm{d}t = 1$$
Thus $$\int_{-\infty}^{\infty}\cos(t)\delta(t)\mathrm{d}t =\int_{-\infty}^{\infty}\cos(0)\delta(t)\mathrm{d}t = \cos(0)\int_{-\infty}^{\infty}\delta(t)\mathrm{d}t = 1\cdot\int_{-\infty}^{\infty}\delta(t)\mathrm{d}t = 1 \cdot 1 = 1 $$
where $\cos(0)=1$.
The step function is defined as $$u(t) = \begin{cases} 0, &t \leq 0 \\ 1, &t \gt 0 \end{cases}$$
Thus squaring the function does not change anything ($u^2(t) = u(t)$).