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

I know from this question that Derivation of Gradshteyn and Ryzhik integral 3.876.1 (in 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:

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

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

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