Say you want to write $\displaystyle \int^{x+ct}_{x-ct} g(x) \ dx = G(x+ct) - G(x-ct) $ (where $\displaystyle G $ is an antiderivative of $\displaystyle g$)

for the odd 2-periodic function $\displaystyle g(x)=\left\{\begin{array}{ccc}-10,&\mbox{ if }

-1 \le x<0 \\10, & \mbox{ if } 0 \le x <1 \\g(x+2L) & \text{otherwise}\end{array}\right.$

then would G(x) be

$\displaystyle G(x)=\left\{\begin{array}{ccc}-\int_{-1}^{x} 10 \ dx = -10x-10 \,&\mbox{ if }

-1 \le x<0 \\ -\int_{-1}^{0} 10 \ dx + \int_{0}^{x}10 \ dx = -10 + 10x, & \mbox{ if } 0 \le x <1 \\F(x+2L) & \text{otherwise}\end{array}\right.$ ?

You couldn't just write

$\displaystyle G(x)=\left\{\begin{array}{ccc}-10x\,&\mbox{ if }

-1 \le x<0 \\ 10x, & \mbox{ if } 0 \le x <1 \\F(x+2L) & \text{otherwise}\end{array}\right.$

could you?