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

3.5 Hyperbolic Functions

3.51 Hyperbolic functions

3.511.

1.¹² 

${\displaystyle {\int }_0^{\infty }\frac{\text{d}x}{\text{cosh }x}=\frac{\pi }{2}}$

2.¹² 

$\begin{array}{ll}{\displaystyle {\int }_0^{\infty }\frac{\text{sinh}ax}{\text{sinh}x}\text{d}x=\frac{\pi }{2}\text{tan}\left(\frac{\pi a}{2}\right)}\hfill & \left[\left|a\right|<1\right]\hfill \end{array}$

BI (27)(10)a

3.¹² 

$\begin{array}{ll}{\displaystyle {\int }_0^{\infty }\frac{\text{sinh}ax}{\text{cosh}x}\text{d}x=\frac{\pi }{2}\text{sec}\left(\frac{\pi a}{2}\right)-\beta \left(\frac{a+1}{2}\right)}\hfill & \left[\left|a\right|<1\right]\hfill \end{array}$

GW (351)(3b)

4.¹² 

$\begin{array}{ll}{\displaystyle {\int }_0^{\infty }\frac{\text{cosh}ax}{\text{cosh}x}\text{d}x=\frac{\pi }{2}\text{sec}\left(\frac{\pi a}{2}\right)}\hfill & \left[\left|a\right|<1\right]\hfill \end{array}$

BI (4)(14)a

5.¹² 

$\begin{array}{ll}{\displaystyle {\int }_0^{\infty }\frac{\text{sinh}ax\text{cosh}bx}{\text{sinh}x}\text{d}x=\frac{\pi }{2}\frac{\text{sin}\pi a}{\text{cos}\pi a+\text{cos}\pi b}}\hfill & \left[\left|a\right|+\left|b\right|<1\right]\hfill \end{array}$

BI (27)(11)

6.^{12 ...}