It seems you're working on homework problems related to wavefunctions and probability densities. Here's an outline of the main tasks:

1. Normalization of Wavefunctions: For each wavefunction, you're asked to find the constant $A$ to ensure that the total probability is 1.

2. Expectation Values: You need to compute the expectation values $\langle x \rangle$ and $\langle x^2 \rangle$ for the various wavefunctions.

3. Standard Deviation: Calculate the standard deviation $\sigma$, using the expectation values.

Here are some key approaches:

1. Normalization of the Wavefunction

You need to set up the integral of the probability density (squared wavefunction) over all space and set it equal to 1: $\int_{-\infty}^{+\infty} |\psi(x)|^2 dx = 1$ This will allow you to solve for $A$.

2. Expectation Values

The expectation value of $x$ is given by: $\langle x \rangle = \int_{-\infty}^{+\infty} x |\psi(x)|^2 dx$ Similarly, for $x^2$: $\langle x^2 \rangle = \int_{-\infty}^{+\infty} x^2 |\psi(x)|^2 dx$

3. Standard Deviation

The standard deviation is found using: $\sigma = \sqrt{\langle x^2 \rangle - \langle x \rangle^2}$

Would you like help with a specific problem from the list, such as normalizing one of the wavefunctions, or calculating an expectation value?

Here are five questions to explore further:

1. How do you approach the normalization of a wavefunction with boundaries, such as $\psi(x)$ defined over $[0, a]$?
2. How is the probability density related to physical measurements of the particle's position?
3. What is the importance of the standard deviation $\sigma$ in quantum mechanics?
4. How would you calculate the expectation value of momentum, $\langle p \rangle$, given the wavefunction $\psi(x)$?
5. How do boundary conditions affect the behavior of a wavefunction?

Tip: When solving for the normalization constant $A$, remember to use symmetry if applicable—it can simplify the integrals!