The integral of $x^x$ with respect to $x$, i.e.,

$\int x^x \, dx$

does not have a closed-form solution in terms of elementary functions (functions like polynomials, exponentials, logarithms, etc.). This integral is classified as a non-elementary integral.

However, it can be expressed in terms of more advanced mathematical functions or solved numerically for specific values of $x$.

Steps:

1. This integral does not simplify easily like a polynomial or basic transcendental function.
2. Techniques such as integration by parts or substitution don't simplify this integral either.
3. Special functions or numerical methods are required to evaluate this for specific values of $x$.

Would you like to know how to solve this numerically or explore special functions?

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Here are 5 related questions to expand your understanding:

1. How are elementary and non-elementary integrals different?
2. What are special functions in mathematics, and when are they used?
3. Can $x^x$ be expressed using logarithms or other simpler functions?
4. How can numerical methods be applied to compute complex integrals like $x^x$?
5. What role does symbolic integration software play in handling non-elementary integrals?

Tip: When faced with a complex integral, checking for special functions or numerical methods is a good strategy for finding approximate solutions.