# How do you integrate $9^{\sin x}$?

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?

• It means it cannot be expressed in terms of standard mathematical functions.
– LHF
Mar 6, 2020 at 12:16
• 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. Mar 6, 2020 at 16:06

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.