Normalization of wave function (Griffiths QM, 2.5)

[Saraphim]

Homework Statement

A particle in the infinite square well has its initial wave function an even mixture of the first two stationary states:

$\Psi(x,0) = A\left[ \psi_1(x) + \psi_2(x) \right]$

Normalize $\Psi(x,0)$. Exploit the orthonormality of $\psi_1$ and $\psi_2$

Homework Equations

$\psi_n(x) = \sqrt{\frac{2}{a}} \sin \left( \frac{n\pi}{a}x\right)$,
where a is the width of the infinite square well.

The Attempt at a Solution

I've managed to eliminate the orthogonal parts of my integral, so I'm now left with
$|A|^2 \int |\psi_1|^2 + |\psi_2|^2 dx = 1$

I have the feeling that I now have to exploit the fact that they are both normalized, but why is that so? What's the logic here?

EDIT: I had written a wrong expression for $\psi_n$. Sorry! :(

 

[Saraphim]

Solution: The expression for $\psi_n$ is already normalized. I should have realized this. Therefore, the integral yields 2 over the interval