I have two functions

I need to write S(x) without the integral sign.

My attempt

I started by trying to find the primitive functions for each part

1.a

2.a

3.a

Now what confuses me a bit is what integrating from 0 to x actually means in this case. Does it just mean that I integrate for the defined interval in each case? Or do I actually calculate things like 1.b here:

1.b

I am also supposed to plot S(x), so I would like to grasp how it works/what it looks like. Right now I just cannot get my head around this.

Any help appreciated.