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I believe this integral $\int_0^a \frac{\cos(ux)}{\sqrt{a^2-x^2}}\mathrm dx$

can not be computed exactly. However is there a method or transformation to express this integral in terms of the cosine integral or similar? I am referring to the integrals here.

$a$ is real number; with the change of variable this integral becomes

$ \int_0^a\cos(u\sin t) \ \mathrm dt $ with $ x=a\sin t, $ So, the new integral is $ \int_0^{\pi /2}\cos(ua\sin t) \ \mathrm dt $

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    aja, thanks what bessel function if possible :) thanks again2012-08-08

1 Answers 1

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From

$\int_0^a \frac{\cos(ux)}{\sqrt{a^2-x^2}}\mathrm dx$

you were able to transform it into

$\int_0^{\pi/2}\cos(au\sin\,t)\mathrm dt$

which is expressible in terms of the Anger function $\mathscr{J}_\nu(z)$, which is equivalent to the more familiar Bessel function of the first kind $J_\nu(z)$ for integer orders:

$\int_0^{\pi/2}\cos(au\sin\,t)\mathrm dt=\frac12\int_0^\pi\cos(au\sin\,t)\mathrm dt=\frac{\pi}{2}J_0(au)$

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    @Jose, note the limits. :)2012-08-09