Regarding Kittel's solid state physics 167 page

[benz994]

Homework Statement

In Kittel's 'Introduction to solid state physics' (8th ed.), on page 167, it says "The wave functions at the Brillouin zone boundary $k=\pi/a$ are $\sqrt{2} cos (\pi x/a)$ and $\sqrt{2} sin(\pi x/a)$, normalized over unit length of line."
Here I cannot understand what is the meaning of "normalized over unit length of line".
Does that mean that, when I integrate the square of the wave function from 0 to 1, the result should be 1? But
$$ \int_0^1 2cos^2 \frac{\pi x}{a} dx = \int_0^1 (1+cos \frac{2\pi x}{a})dx $$
$$ = 1+\frac{a}{2\pi} sin\frac{2\pi}{a} \neq 1. $$
Please help me find out what is wrong in my reasoning.
Thanks.

 

[voko]

Is $a$ given, or is it something you can specify at will? In the latter case, can you find $a$ that would satisfy the requirement?

 

[benz994]

Thank you for the responce.
I think $a$ is a given quantity. $a$ is the size of the unit cell in one-dimensional lattice.
The above equation is satisfied only when $a=1$.

 

[voko]

The above equation is satisfied only when $a=1$.

No, not only. The sine is a periodic function.