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.
2026-03-31 21:11:50.1774991510
Integral of cosine multiplied by zeroth-order Bessel function
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I found the answer from Gradshteyn and Ryzhik's book, 7th edition section 6.677 the 6th equation. $$\int_0^1 \cos(ax)\ J_0\left(b\sqrt{1-x^2}\right)\ \mathrm{d}x = \frac{\sin\left(\sqrt{a^2+b^2}\right)}{\sqrt{a^2+b^2}}$$