The output $y(t)$ is the convolution of input $x(t)$ with impulse response $h(t)$: $$ y(t) = h(t) * x(t) $$
This is a linear, time invariant system.
What is the output $y(t)$ in real form when you have input $x(t) = \cos(2 \pi t)$, $-\infty < t < \infty$ and frequency response response $h(t) = u(t) - u(t - 1/2)$? $u(t)$ is the unit step function.
I tried: \begin{align*} &\, \int_{-\infty}^\infty (u(\tau) - u(\tau - 1/2)) * \cos(2 \pi (t - \tau) ) \,\text{d} \tau \\ =&\, \int_{0}^{1/2} \cos(2 \pi (t - \tau)) \,\text{d}\tau = \left[ \frac{-\sin(2 \pi t - 2 \pi \tau)} {2\pi} \right]_0^{1/2} \\ &= \frac{- \sin(2 \pi t - \pi)} {2 \pi} -\frac{-\sin(2 \pi t)}{2\pi} = - \frac{\sin(2 \pi t - \pi)}{2\pi} + \frac{\sin(2 \pi t)}{2\pi} \\ &= \frac{\sin(2 \pi t)}{2\pi} + \frac{\sin(2 \pi t)}{2\pi} = \frac{\sin(2 \pi t)}{\pi} \end{align*}