Integral of cosine multiplied by zeroth-order Bessel function I am looking for the result of the integral below,
$$ \int_0^1 \cos(ax)\ J_0(b\sqrt{1-x^2})\ \mathrm{d}x$$
where $J_0(x)$ is the zeroth order Bessel function of the first kind. Variables $a$ and $b$ are known and real.

I found the answer from Gradshteyn and Ryzhik's book, 7th edition section 6.677 the 6th equation.
$$\int_0^1 \cos(ax)\ J_0\left(b\sqrt{1-x^2}\right)\ \mathrm{d}x = \frac{\sin\left(\sqrt{a^2+b^2}\right)}{\sqrt{a^2+b^2}}$$
