I've come across a problem that involves something very close to: $$\int(\cos{x})^xdx$$ and I have no clue as to how to proceed with any kind of analysis for this type of equation. It occurred to me that complex analysis might lend some clarity, maybe the rewrite: $$\int(\cos{x})^xdx=\int\dfrac{(e^{i*x}+e^{-i*x})^x}{2^x}dx$$ would point me in the correct direction, but analytically I'm still stumped here.
Does anyone have any knowledge of this family of equations? The integral probably can't be analytically evaluated with the maths we have today, but any information about the function type would be appreciated.
Yes, $\cos(x)^x$ is not hard to "explore". I don't know if it's been the subject of any published work, other than OEIS sequence A215347. It satisfies, for example, the differential equation $$ y y'' = (y')^2 - (2 \tan(x) + x \sec^2(x)) y^2$$ and it has the Maclaurin series
$$ y = 1 - \dfrac{x^3}{2} - \dfrac{x^5}{12} + \dfrac{x^6}{8} - \dfrac{x^7}{45} + \dfrac{x^8}{24} - \dfrac{139}{5040} x^9 + \ldots $$
As for being defined, $(\cos x)^x = \exp(x \ln \cos x)$. This will be multivalued on the complex plane with logarithmic branch points where $\cos x = 0$.