I have two functions

$\displaystyle f(t)=\left\{\begin{array}{ccc}\sqrt{t^2+2\pi t + \pi^2} &\mbox{ if }& t < -\pi \\ \cos{\left(\frac{\pi}{2} - t\right)} & \mbox{ if }& -\pi\leq t \leq \frac{\pi}{2}\\ 1 &\mbox{ if }& t > \frac{\pi}{2} \end{array}\right.$

$\displaystyle S(x)=\int_0 ^x f(t) dt \ \ \ \ -\infty < x < \infty$

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

$\displaystyle \int \sqrt{t^2 + 2\pi t + \pi^2} dt = \int \sqrt{(x+\pi)^2}= \frac{1}{2}(t+\pi)^2 +C$

2.a

$\displaystyle \int \cos{\left(\frac{\pi}{2}-t\right)} dt=\int \sin{t} dt = -\cos{t} + C$

3.a

$\displaystyle \int 1 dt = t + C$

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

$\displaystyle \frac{1}{2}\left[(t+\pi)^2\right]_0 ^x = \frac{1}{2}\left( (x+\pi)^2 - \pi^2 \right)=\frac{1}{2}(x^2+2\pi x)$

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.