What is the integral to $\int \frac{\sin \left( xt\sqrt{t^{2}+a^{2}}\right)}{t} \, dt$

Question

$\int_{0}^{\infty} \frac{\sin \left( x\sqrt{t^{2}+a^{2}}\right)}{\sqrt{t^{2}+a^{2}}} \, dt = \frac{\pi}{2} J_{0}(ax), \quad (a>0, \ x>0)$

What are the integrals to:

$\int_{0}^{\infty} \frac{\sin \left( xt\sqrt{t^{2}+a^{2}}\right)}{t} \, dt = ?, \quad (a>0, \ x>0)$

$\int_{n}^{m} \frac{\sin \left( xt\sqrt{t^{2}+a^{2}}\right)}{t} \, dt = ?, \quad (a>0, \ x>0, \ m > n > 0)$

@MaximilianJanisch Please don't add a newly created tag to many questions at once. And whenever you want to create a new tag, it's preferable to post first in the Tag management thread to see what other users think of the tag. Alternatively, there is also a dedicated chatroom. — Arnaud D., Sep 06 '19 at 15:30

score 0 · Answer 1 · answered Apr 05 '18 at 20:01

0

hint consider first and second derivation with respect to $x$

answered Apr 05 '18 at 20:01

openspace

1 Answers1