How do you integrate cos(cos x)?

Question

I started with the definition / alternative form of $\cos(x) := \frac{1}{2} \cdot (e^{-ix} + e^{ix})$ and nested it $\frac{1}{2} \cdot (e^{-\frac{1}{2}i(e^{-ix} + e^{ix})} + e^{\frac{1}{2}i(e^{-ix} + e^{ix})})$

Well and now I'm a little bit overwhelmed what to do next and how to integrate it. Is there a trick?

I don't think this function can be integrated in terms of elementary functions. — DonAntonio, Feb 23 '16 at 08:42
How do you integrate $\cos(\cos x)$ ? - Like this... :-$)$ — Lucian, Feb 23 '16 at 08:42
What makes you think such an integral even exists? I'm about 99% sure there is no integral in elementary functions. — A. Thomas Yerger, Feb 23 '16 at 08:43

score 3 · Accepted Answer · answered Feb 23 '16 at 09:17

You don't (in terms of elementary functions).

Unless a power series will do...

$$\cos (1) (x-\pi ) +\frac{1}{6} \sin (1)(x-\pi )^3 + \left(-\frac{\sin (1)}{120}-\frac{\cos (1)}{40}\right)(x-\pi )^5 + \left(\frac{\cos (1)}{336}-\frac{\sin (1)}{360}\right)(x-\pi )^7 + \frac{209 \sin (1)+42 \cos(1)}{362880}(x-\pi )^9 + \dots $$

Warning: the rate of convergence is not particularly good...

How do you integrate cos(cos x)?

1 Answers1