The undergraduate calendar lists one of Math 1A03, 1LS3 or 1X03 as a prereq for Physics 1B03 (see
here). So yes, you need to take one of them. It doesn't really matter which one, though 1A03 will probably make you more familiar with integration than 1LS3.
And yes, you need to know differentiation as well as basic integration for Physics 1B03 (I took Physics 1B03 last year).