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Find integral: x(x)dx\int x^{(x)}\,dx

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Q. Find integral: x(x)dx\int x^{(x)}\,dx
  1. Recognize Problem Complexity: Recognize that the integral of xxx^{x} with respect to xx is not an elementary function and cannot be expressed in terms of elementary functions.
  2. Use Sophomore's Dream Function: Since the integral cannot be expressed in terms of elementary functions, we cannot find a closed-form solution. However, we can express the integral using a special function known as the Sophomore's dream function, which is not commonly used in elementary calculus.
  3. Define Sophomore's Dream Function: The Sophomore's dream function is defined as the integral of xx raised to the xx power, but it does not have a simple expression. It is usually denoted as extint(xxextdx) ext{int}(x^x ext{d}x) and is a known function without a simple antiderivative.
  4. Conclude with Special Function: Since we cannot find a simple antiderivative, we conclude that the integral of xxx^{x} with respect to xx is not expressible in terms of elementary functions and can only be represented using the special function notation or numerical methods.

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