Solution of the Wave Equation

Hi,

I'm having a bit of trouble understanding D'Alembert's solution to the wave equation.

**The Problem**

Or, in short notation:

Find subject to the initial conditions:

**The Solution (the part I understand)**

The first stage is to express the equation in characteristic coordinates, which, once performed, gives:

Here, and .

Integrating the equation w.r.t both variables gives us the simple general solution:

Or, back in cartesian coordinates:

(1)

Now, differentiating this, we get:

(2)

Applying the initial conditions to equations (1) and (2), we get:

(3)

(4)

**The Solution (part I don't understand)**

Now, the book I am using then makes the following statement:

If we integrate equation (4), we get:

To be honest, this is the only part I don't understand. Where do the other terms which results from the integration disappear to? i.e., by my working, I would get:

How do they manage to conclude that ??