To integrate $e^{0.45t^2}$, we can approach the problem, but this integral does not have a solution in terms of elementary functions. However, it is related to the Gaussian integral and often expressed in terms of the error function (erf).

Solution:

The integral of $e^{kt^2}$, where $k$ is a constant, typically requires special functions to represent the result. Unfortunately, the constant in front of $t^2$, here $0.45$, doesn't allow for a straightforward antiderivative.

Thus, we can express it as:

$\int e^{0.45 t^2} \, dt$

This is closely related to the error function for integrals of the form $\int e^{ax^2} dx$, but it needs to be solved numerically or left in terms of a special function.

If we look for a numerical solution, that would require defining limits.

Would you like more details on using numerical methods or working with error functions?

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Here are 5 questions to expand on this:

1. Would you like to evaluate this integral with specific limits for a numerical result?
2. Are you familiar with the error function and how it's used in integrals?
3. Do you need the integral evaluated in terms of a power series expansion?
4. Would you like an example of a related integral that does have an elementary solution?
5. How does this integral fit into your larger problem?

Tip: Special functions like the error function often arise in integrals involving exponentials and squares, and are frequently used in probability and statistics.