For any function f continuous on $\,(-\infty\,,\,\infty)\,$:
$$g(x) = \int_0^x f(t)\,dt$$
$$h(x) = \int_0^x (x-t)f(t)\,dt$$
$$w(x) = \int_0^x f(t)\sin(x-t)\,dt$$
Show that
$$h(x) = \int_0^x g(u)\, du$$
and
$$ \frac{d^2w}{dx^2} + w = f(x) w(0)=0\,\,,\,\text{and}\,\, w'(0) = 0$$