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

${\displaystyle {\int }_0^{\infty }cos{\left(ax-\frac{b}{x}\right)}^2}\frac{\text{d}x}{x^2}=\frac{\sqrt{2\pi }}{4b}$

  BI (179)(14)a

[a >; 0, b > 0]

3.875

1. 

${\displaystyle {\int }_u^{\infty }\frac{xsin\left(p\sqrt{x^2-u^2}\right)}{x^2+a^2}}cosbx\text{d}x=\frac{\pi }{2}\exp \left(-p\sqrt{a^2+u^2}\right)coshab$

  ET I 27(39)

[0 < b < p]

2. 

${\displaystyle {\int }_u^{\infty }\frac{xsin\left(p\sqrt{x^2-u^2}\right)}{a^2+x^2-u^2}}cosbx\text{d}x=\frac{\pi }{2}{e}^{-ap}\left(b\sqrt{u^2-a^2}\right)$

  ET I 27(38)

[0 < b < p, a < 0]

3.¹² 

${\displaystyle {\int }_0^{\infty }\frac{sin\left(p\sqrt{a^2+x^2}\right)}{{\left(a^2+x^2\right)}^2}}cosbx\text{d}x=\frac{\pi p}{2a}{e}^{-ap}$

  ET I 26(29)

[0 < p < b, b >; a >; 0]

4.^* 

${\displaystyle {\int }_0^{\infty }\frac{sin\left(p\sqrt{x^2+a^2}\right)}{{\left(x^2+a^2\right)}^2}}cos\left(bx\right)\text{d}$