Divide the integral with sums in two integrals?

How can I divide (generically) an integral of the form

To integrate[f[x] + g[x], {x, l, h}]

in

To integrate[f[x], {x, l, h}]+ Integrate[g[g[g[g[x], {x, l, h}] ?

I thought that Develop or A part could do it, but apparently they do not do it. Thank you.