Wave funtion and normalisation constant

[rayman123]

Homework Statement

find the normalisation constant (A)

Homework Equations

wave functcion
$$\psi (x,t)= A[3sin(\frac{\pi x}{L})+2sin(\frac{2\pi x}{L}]$$

The Attempt at a Solution

$$A^2\int_{0}^{L}[\psi(x,t)]^2dx=1$$
$$A^2\int_{0}^{L}[9sin^2(\frac{\pi x}{L})+12sin(\frac{\pi x}{L})sin(\frac{2\pi x}{L})+4sin^2(\frac{2\pi x}{L})}]dx$$

Homework Statement

i try to solve each integral separately
I have started with the first one and i got

$$A^2\int_{0}^{L}[9sin^2(\frac{\pi x}{L})]dx= A^2\cdot \frac{9L}{ \pi}\int_{0}^{\pi}sin^2tdt=A^2\cdot \frac{9L}{\pi}[\frac{1}{2}t+\frac{sin(2t)}{4}]\right]_{0}^{\pi}=A^2\cdot \frac{9}{2}L$$
is it correct so far?

from the last integral i got
$$A^2\frac{L}{\2\pi}\int_{0}^{2\pi}4sin^2(2t)dt=A^22L$$

i don't have a good idea for solving the second intregral...

 

[thebigstar25]

use the fact that, 2 sin A sin B = - cos (A + B) + cos (A - B) ..