Table of Integrals, Series, and Products, 8th Edition

6.69–6.74 Combinations of Bessel and trigonometric functions and powers

6.691

$\begin{array}{cc}{\displaystyle {\int }_0^{\infty }xsin\left(bx\right){\mathit{K}}_0\left(ax\right)\text{d}x=\frac{\pi b}{2}{\left(a^2+b^2\right)}^{-\frac{3}{2}}}& \left[\begin{array}{cc}Rea>0,& b>0\end{array}\right]\end{array}$

  ET I 105(47)

6.692

1. 

$\begin{array}{cc}{\displaystyle {\int }_0^{\infty }x{\mathit{K}}_{\nu }\left(ax\right)I_{\nu }\left(bx\right)sin\left(cx\right)\text{d}x=-\frac{1}{2}{\left(ab\right)}^{-\frac{3}{2}}c{\left(u^2-1\right)}^{-\frac{1}{2}}{Q}_{\nu -\frac{1}{2}}^1\left(u\right),}& u={\left(2ab\right)}^{-1}\left(a^2+b^2+c^2\right)\end{array}$

  ET I 106(54)

$\left[\begin{array}{ccc}Rea>\left|Reb\right|,& c>0,& Re\nu >-\frac{3}{2}\end{array}\right]$

2. 

${\displaystyle {\int }_0^{\infty }x{\mathit{K}}_v\left(ax\right){\mathit{K}}_{\nu }\left(bx\right)sin\left(cx\right)\text{d}x=\frac{\pi }{4}{\left(ab\right)}^{-\frac{3}{2}}c{\left(u^2-1\right)}^{-\frac{1}{2}}\Gamma \left(\frac{3}{2}+\nu \right)\Gamma \left(\frac{3}{2}-\nu \right)P_{\nu -\frac{1}{2}}^{-1}\left(u\right)}$

  ET I 107(61) ...