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I am looking through rules/tricks for integration, and there are a lot... but I can't seem to find one specifically applicable to the general form $$\int_a^b f(x)\,f(f(x)) \,\mathrm dx$$

Is there a nice rule or trick that would generally apply in this situation?

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How did this come up? I am guessing that $f(x) = e^x$, in which case you are generalizing incorrectly; recall that $f(x) = f'(x)$ in this case... –  Qiaochu Yuan Sep 26 '11 at 16:07
    
@Qiaochu: f(x) does not equal $e^x$ -- that's why I'm asking about a general rule. Mostly, I am new to this stuff, and it seems like there is a near endless list of tricks, and I was just wondering if I missed one.. –  Angada Sep 26 '11 at 16:27
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@Angada: there's no reason to expect a rule for integrals of this kind. The only general rules with wide applicability that I can really think of are $u$-substitution and integration by parts. –  Qiaochu Yuan Sep 26 '11 at 16:29
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3 Answers

up vote 5 down vote accepted

Not in general. If, say, $f(x)=\ln\,x$, one requires a nonelementary function to represent the integral...

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No, but there is a rule for integrating $f'(x)\,f'(f(x))$.

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what is rule of it? –  nim Nov 21 '13 at 13:58
    
The derivative of $f(f(x))$ is what I wrote. So, the integral of what I wrote is $f(f(x)) +C$. –  GEdgar Nov 21 '13 at 14:05
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I suggest integration by parts where $u=f(x)$ and $dv=f(f(x))dx$

$du=f'(x)dx$ and $v=\int f(f(x))dx$ , this integral we may evaluate as $v=\int f(x)dx \int f(x)dx$

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