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When learning integration there are rules for different combinations of functions. For $f(x)\pm g(x)$ you apply linearity, $f(g(x))$ can be handled by substitution rule and $f(x)g(x)$ is integration by parts. But there seems to be nothing for the case where a function is raised to the power of another function.

$$\int f(x)^{g(x)} dx$$

Is there any rule or technique you can apply to the integral above or is it just a dead end?

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    $\begingroup$ No rule for this. $\endgroup$
    – Ethan Bolker
    Jun 6, 2022 at 17:34
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    $\begingroup$ Even if $f$ or $g$ is constant but not both, there's no standard rule. $\endgroup$
    – J.G.
    Jun 6, 2022 at 17:39
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    $\begingroup$ No. Even with simple functions $f$ and $g$, the integral may have no closed form : for example, $\int x^x dx$ has no closed form. $\endgroup$
    – TheSilverDoe
    Jun 6, 2022 at 17:45
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    $\begingroup$ To be fair, composition and multiplication can already generate integrals with no closed form even if the constituent functions do have elementary antiderivatives. So exponentiation is not an outlier here (indeed this shows how special addition and constant multiplication are). $\endgroup$
    – Greg Martin
    Jun 6, 2022 at 18:11
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    $\begingroup$ Just to make it perfectly clear, even if other comments already observed, composition not always can be handled by substitution ($\sin(x^2)$) and product not always can be handled by integration by parts ($\sin(x)\log(x)$). $\endgroup$
    – jjagmath
    Jun 6, 2022 at 19:10

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In the (rare) case where the integral can be calculated, the best way to begin is probably write $f(x)^{g(x)}$ as $e^{g(x) \log(f(x))}$. After that there's no much that can be done in general.

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