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

[dirk_mec1]

Homework Statement

<br /> \int_{0}^{\pi} \cos ( \sin x ) \mbox{d}x<br />

The Attempt at a Solution

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

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.