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Because integrals are not something to be "solved" by default, but a tool of mathematical expression in their own right. Can you express $\sqrt{2}$ as something else? Probably nothing simpler. You can approximate it, but that's about it. Likewise, why should you expect to be able to express some arbitrary integral, say $$\int_{0}^{1} e^{\sin x}\ dx$$ as ...


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I would point out that integration is the reverse of differentiation, a pretty straightforward process. But often times the reverse of an easy algorithm is surprisingly hard! Solving a linear system of equations vs. plugging a value into linear equations Factoring vs. expanding (polynomial and prime factoring) Finding the inverse of a function at a point vs....


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