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I understand how you would differentiate it, but how would you integrate it? After plugging it into integral calculator, it says that it is non-elemetary, what does this mean too?

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    $\begingroup$ It means it cannot be expressed in terms of standard mathematical functions. $\endgroup$
    – LHF
    Mar 6, 2020 at 12:16
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    $\begingroup$ This is a useful lesson for you. Most formulas you can write down have only non-elementary integrals. Indefinite integrals with closed-form answers are the exceptions, not the rule. $\endgroup$
    – GEdgar
    Mar 6, 2020 at 16:06

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It does not have any closed-form. What we know is $$\int _{x=0}^{2\pi} 9^{\sin(x)} \mathrm{d} x=\int _{x=0}^{2\pi} e^{\ln(9)\sin(x)} \mathrm{d} x=\mathrm{I}_0(\ln(9)),$$ where $\mathrm{I}_0(\cdot)$ is the modified Bessel function of the first kind.

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