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

## 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?

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Dick
Homework Helper

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

So n = 0 here:

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

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