How would we find $ \int e ^{x(e^x+1)} dx $ ?
If the integral cannot be expressed in elementary functions, then how can we prove that it cannot be expressed in elementary functions?
2026-03-27 17:42:10.1774633330
How would we find $ \int e ^{x(e^x+1)} dx $?
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By using $e^{x}=u , e^{x}dx=udx=du $ we reach to this one: $$ \int e^{x(e^{x}+1)}dx=\int e^{xe^{x}}.e^{x}dx=\int e^{xu}.u\frac{du}{u}=\int (e^{x})^{u}du=\int u^{u}du$$
I don't continue for integration of this equation. Please refer to these pages:
Finding $\int x^xdx$
$\int x^x\,dx$ - What is it, and why?
https://www.quora.com/What-is-int-x-x-dx