I am trying to find a series expansion of
$\displaystyle f(x) = \int_{0}^{\pi/2}\cos(x\cos{\theta})d\theta \tag*{}$
To do that, I tried to use the power series of $\cos{x}$. We have
$\displaystyle f(x) = \int_{0}^{\pi/2}\sum_{n=0}^\infty (-1)^n \frac{(x\cos{\theta})^{2n}}{(2n)!} d\theta \tag*{}$
Now, I would like to exchange summation and integral, but I don't know how to prove the uniform convergence of the summation.
My friend told me that it is uniformly convergent since the power series of cos has a radius of convergence $\infty$, but I don't know how to fill a gap to obtain uniform convergence.