In an infinite square well, how do you find the probability of the particle?

Question

If my initial wavefunction is given by
psi={Ax(a-x), 0 0, elsewhere

and A=SQRT(30/a)

How would I find what probability of the particle described by this wavefunction will be found in the region -0.5a
I know the probability between -0.5

Answer 1

Normally you should find the probability by integrating
probablitlies of the particle

dp = |Ψ|² dx = ΨΨ* dx
P = int [ΨΨ* dx] = A² int [ x²(x - a)² dx ]

This integral can be easily calculated, but in this
particular case this would not be necessary.
Since Ψ(x) is symmetric with respect to x=a/2,
P[-inf ... a/2] = 1/2.