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

1. Apr 17, 2009

### dirk_mec1

1. The problem statement, all variables and given/known data

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

3. 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?

2. Apr 17, 2009

### Dick

Re: integral

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

3. Apr 17, 2009

### dirk_mec1

Re: integral

So n = 0 here:

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

4. Apr 17, 2009

### Dick

Re: integral

Sure. n=0, x=1.