How is
$$ \int {f(x)}^{g(x)} dx $$
neatly expressed in terms of:
$$ \int f(x) dx {,\,} \int g(x) dx ? $$
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3It's (provably) not possible. You can't even do this for $f(x) = 2$ (constant function) and $g(x) = x^{2}.$ – Dave L. Renfro Apr 28 '16 at 19:08
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Why Is it not a pity ? – Narasimham Apr 28 '16 at 19:10
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See the similar question on integral of a product: http://math.stackexchange.com/questions/50671 – GEdgar Apr 28 '16 at 19:12
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1@Narasimham Yes, it is a pity. Bad behavior under most non-linear operations is the reason why integrals are hard to calculate. – Apr 28 '16 at 19:13
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1As for why, here's a proof for $f(x) = x$ and $g(x) = x$: How to determine with certainty that a function has no elementary antiderivative? It's way past elementary calculus level, however. – Dave L. Renfro Apr 28 '16 at 19:19
1 Answers
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There is no neat expression. Compare $$ \int (1-x^2) \;dx = x-\frac{x^3}{3}+C \\ \int \left(-\frac{1}{2}\right)dx = -\frac{x}{2}+C $$ with $$ \int (1-x^2)^{-1/2}\;dx = \arcsin x +C $$
GEdgar
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So it is impossible to generate the arcsin infinite series using $-x/2, x -x^3/3$, right? – Narasimham Apr 30 '16 at 01:45