Analytical Solution for the Convolution of Signal with a Box Filter

I have an exercise in which I am trying to filter an input signal $y(x) = \sin(x)$. Ideally, I would like to apply a box filter to this signal.

Previously, I successfully convolved the input signal $y(x)$ with a decaying response $h(x) = e^{-x}$.

I did so by the following the definition of convolution (e.g., integrating $\int_0^t\sin(x')e^{-(x-x')}\mathrm{d}x'$ and computing a damped sinusoidal signal.

My box filter is given by $\frac{1}{\Delta}$ for$|x-\xi| \leq \frac{\Delta}{2}$ and 0 elsewhere, where $\Delta$ is the filter width. I understand that a box filter is a local average, and I can implement this numerically, but I do not understand how to analytically integrate this as I did with the damped exponential 'filter'.

I tried to take the Fourier transform of $y(x)$ and $h(x)$ and multiplying them in Fourier space, but I could not figure out how to do so.

Thanks for any help.