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Can every integration and differentiation on $\mathbb{R}^2$ be determined exactly?

I am curious of this, because I know that there are some integration and differentiation that do not yet have a way to solve them.

However, I also never heard of any theorem that state that there are some integrals and differentiations that cannot be solved.

So, is there any theorem?

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I'm sorry, but your question isn't clear. Are you asking if every function defined from a subset of $\mathbb{R}^2$ to $\mathbb{R}^2$ is integrable and differentiable, or if all derivatives/integrales can be expressed in terms of simple functions or about function infinitely-differentiable/integrable? Please clarify what you mean. –  andreas.vitikan Aug 24 '12 at 9:24
Do you mean like $$\int e^{-x^2}dx$$ and why is this tagged [logic] anyway? –  Asaf Karagila Aug 24 '12 at 9:27
The question is unclear, but perhaps you mean How can you prove that a function has no closed form integral? –  Rahul Aug 24 '12 at 9:27
You have to clarify what you mean by "determined exactly" and whether you are talking about integral with number values or integrals like the one Asaf mentions in his comment. –  Kaveh Aug 24 '12 at 13:34
sorry, guys. Ehat Rahul Narain says is what I intended to ask. I think one should close this question. Thanks guys. –  Tao Mao Aug 25 '12 at 13:38

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