Periodic functions and properties of integration This is from a book in Fourier series. 
I don't understand how the integral can be split as in equality 2 and 3. 
Which properties of integration justify that?
$F(x)$ and $f(x)$ are two functions with the periodicity $2l$. The integral $F(x)=\int_{-l}^x f(t) \, \mathrm dt$ is equal to zero at $x=-l$ and at $x=l$.
\begin{align*}
F(x+2l)= \int_{-l}^{x+2l} f(t) \,\mathrm dt &= \underbrace{  \int_{-l}^x f(t) \, \mathrm dt+ \int_x^{x+2l}f(t)\, \mathrm dt }_\text{How?}\\
&= \underbrace{ \int_{-l}^x f(t)\, \mathrm dt + \int_0^{2l} f(t) \, dt}_\text{How?} \\
&=\int_{-l}^x f(t) \, \mathrm dt= F(x).
\end{align*}

Let us consider the function f (x)=cos x.
At  x=-phi and+phi f (x) is zero.
 Here,l=phi and let us consider x=phi/6.
When considered the area of f (x) from -phi to 13phi/6.
In the region 0 to phi the curve area is positive.
In the region phi to 2phi the curve area is negative.
Now cosider the area from -phi to 13phi/6.we can divide this area as (-phi to 0), (0 to phi), (phi to 2phi), (2phi to 13phi/6).since the areas (0 to phi), (phi to 2phi) are equal and opposite in direction they cancel each other.Now we are left with the areas (-phi to 0) and (2phi to 13phi/6).this is what you got
