I saw a YouTube video that messed up all my intuition. They were calculating the integral of something like -1+x^dx which in the end he solved by taking the limit as dx->0 and he got the solution. But everyone in the comments was saying complex stuff like like “well to be rigorous x^dx can be thought of an element of C*(R,R), the exterior algebra on R”. Huh. When did dx get any different from lim x->0.
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The most important thing to take away is that the integral of x^d^x is not a defined expression and is fundamentally meaningless.
The dx in the context of classic integrals and derivatives is purely symbolic and does not have a definition but *represents* the width of an infinitely small interval. While it can sometimes be convenient to treat the dx like a factor or divisor, that is technically not rigorous and can lead to errors.
Therefore, the question “what is the integral of x^d^x – 1” does not have a proper answer because in the context of calculus exponentiating to dx is not a thing. However, you can still think about what this expression *could* represent *if* you ignore that it is not rigorous, in the same way that I could ask you what the color purple should taste like, even though there is no rigorous answer to that either. [Here](https://math.stackexchange.com/questions/3201425/integrating-a-function-with-mathrm-dx-as-an-exponent) are some interesting answers that one might get using different approaches.
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