Thread: Please help me on understanding the answer of this D'Alembert problem.

This is an example I copy from the book. The book showed the steps of solving and provide the answer. I don't understand the book at all. Below I show the question and the solution from the book. Then I am going to ask my question at the bottom.

Question
Use D'Alembert method to solve the wave equation with boundary and initial value:

Given: f(x)=0 & g(x)=x for 0<x<1. c=1, L=1.


D'Alembert:



f(x) = 0 (1)

Where is the odd 2-period extension of g and G is the antiderivative of
Let . (2)
To complete the solution, we must determine G. We know G on any interval of length 2. Since on the interval (-1,1), we obtain
. . for x in (-1,1) (3) . Hence



My question:
1) How does it jump from (1) to (3)??
2) Is (2) just the first Fundamental Theorem of Calculus where -1 is the lower limit of x on [-1,1]?
3) Where is (3) come from? How are (x+t) and (x-t) change to x and -1 respectively?
I don't even understand the answer of the book!! Can anyone explain to me?
Thanks

Anyone please?

d'Alembert Solution of the Wave Equation