How to integrate cos( sin(x) ) from 0 to pi

[dirk_mec1]

Homework Statement

$$\int_{0}^{\pi} \cos ( \sin x ) \mbox{d}x$$

The Attempt at a Solution

If I use $u = \pi-x$ I get :
$$\int_{0}^{\pi} \cos ( \sin x ) \mbox{d}x = \int_{0}^{\pi} \sin ( \cos x ) \mbox{d}x$$

but then what?

 

[Dick]

That reminds me of a definition of the Bessel function in terms of integrals...

 

[dirk_mec1]

So n = 0 here:

$$J_n(x) = \frac{1}{\pi} \int_{0}^{\pi} \cos (n \tau - x \sin \tau) \,\mathrm{d}\tau.$$

 

[Dick]

So n = 0 here:

$$J_n(x) = \frac{1}{\pi} \int_{0}^{\pi} \cos (n \tau - x \sin \tau) \,\mathrm{d}\tau.$$

Sure. n=0, x=1.