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