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intx^(x)dx

∫xxdx \int x^{x} d x

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Full solution

Q. ∫xxdx \int x^{x} d x
  1. Recognize complexity: Recognize the integral form and complexity.\newlineIntegral: ∫xx dx\int x^x \, dx\newlineThis integral is known for not having a standard elementary form.
  2. Attempt substitution: Attempt substitution or simplification.\newlineLet's try substitution, but it's tricky because xxx^x is not easily simplified using basic substitution methods.
  3. Consider special functions: Consider special functions or series expansion.\newlineThe function xxx^x can be expressed using the exponential function: xx=exln⁡(x)x^x = e^{x \ln(x)}.\newlineHowever, integrating exln⁡(x)e^{x \ln(x)} directly is also complex and doesn't lead to a standard form.
  4. Realize advanced methods: Realize the need for advanced methods or numerical approximation.\newlineThis integral, ∫xx dx\int x^x \, dx, generally requires numerical methods or special functions like the Gamma function for specific bounds, or it remains as an indefinite integral without a closed form.

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