Can you explain the working to this question? Probability - infinite potential well

Question 3, I copied the way to do it from the board last year but don't understand it. Can you explain why there's a subtitle 2 beside the psi? And explain why the working is worked out like this?

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Why $|\Psi|^2$ ?

It has to do with Born-interpretation of wavefunction.

$|\Psi|^2$ is interpreted as the probability density that the quantum particle is at the position x.

Then the probability of the quantum particle is found in the interval of $a \leq x \leq b$ is

$\int_a^b |\Psi(x)|^2 dx$

It seems that the wavefunction has something missing. See the link below.
http://hyperphysics.phy-astr.gsu.edu/hbase/quantum/pbox.html

You may want to ask your classmate for a copy of the solution. It seems that your integration is wrong.

It's just normal integration. Note that there's a 2/a outside one bracket and a 1/2 immediately to its right. So multiplying them (during the integration step) gives 1/a. That's not affected by integration because x is the integration variable (not a).

Regarding the a not cancelling: check your very last step.

General tip, either use decimals or (better) only fractions. Something like '2*pi*(0.25a)/a' is unnecessarily complicated notation. (2*pi*a)/(4*a) is so much easier to clean up in a single step without a mistake. It's also faster to write in an exam.

Thanks, I understand it now. I cleaned up the last line and got 0.25-1/2pisin(0.5pi) and got 0.24.... and not the 0.091... it's supposed to be?

Make sure your calculator is in radian mode.