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

4.62 Double and triple integrals with constant limits

4.620 General formulas

1. 

${\displaystyle {\int }_0^{\pi }\text{d}\omega \text{?}{\displaystyle {\int }_0^{\infty }{f}^{\prime }\text{?}(p\text{?}cosh\text{?}x+q\text{?}\text{cos}\text{?}\omega \text{?}\text{sinh}\text{?}x)\text{?}\text{sinh}\text{?}x\text{?}\text{d}x\text{?}=-\frac{\pi \text{?}\text{sign}\text{?}p}{\sqrt{p^2-q^2}}f\left(\text{sign}\text{?}p\sqrt{p^2-q^2}\right)}}$

  LO III 389

$\left[\begin{array}{ll}{p}^2>q^2,\hfill & \underset{x\to +\infty }{\lim }f\left(x\right)=0\hfill \end{array}\right]$

2. 

${{\displaystyle \int }}_0^{2\pi }\text{d}\omega {{\displaystyle \int }}_0^{\infty }{f}^{\prime }[p\text{?}cosh\text{?}x+(q\text{?}cos\text{?}\omega +r\text{?}sin\text{?}\omega )\text{?}sinh\text{?}x]\text{?}sinh\text{?}x\text{?}\text{d}x=-\frac{2\pi \text{?}sign\text{?}p}{\sqrt{p^2-q^2-r^2}}f(signp\sqrt{p^2-q^2-r^2})$

  LO III 390

$[\text{?}\text{?}\begin{array}{ll}{p}^2>q^2+r^2,\hfill & \underset{x\to +\infty }{\lim }f(x)=0\hfill \end{array}]$

3. 

${\displaystyle {\int }_0^{\pi }{\displaystyle {\int }_0^{\pi }\frac{\text{d}x\text{?}\text{d}y}{}}}$