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

3.3–3.4 Exponential Functions

3.31 Exponential functions

3.310¹¹

$\begin{array}{ll}{\displaystyle {\int }_0^{\infty }{e}^{-px}\text{d}x=\frac{1}{p}}\hfill & \text{[Re}\text{p > 0]}\hfill \end{array}$

3.311

1. 

${\displaystyle {\int }_0^{\infty }\frac{\text{d}x}{1+e^{px}}=\frac{\ln 2}{p}}$

  LO III 284a

2. 

$\begin{array}{ll}{\displaystyle {\int }_0^{\infty }\frac{e^{-\mu x}}{1+e^{-x}}\text{d}x=\beta \left(\mu \right)}\hfill & \text{[}\mathrm{Re}\mu \text{ > 0]}\hfill \end{array}$

  EH I 20(3), ET I 144(7)

3.¹² 

$\begin{array}{ll}{\displaystyle {\int }_{-\infty }^{\infty }\frac{e^{-ax}\text{d}x}{1+e^{-x}}=\frac{\pi }{sin\pi a}}\hfill & \text{[}\mathrm{Re}a\text{ > 0]}\hfill \end{array}$

  (cf. 3.241 2)BI (28)(7)

4. 

$\begin{array}{ll}{\displaystyle {\int }_0^{\infty }\frac{e^{-qx}\text{d}x}{1-a{e}^{-px}}={\displaystyle \sum _{k=0}^{\infty }\frac{a^k}{q+kp}}}\hfill & \text{[0 < a < 1]}\hfill \end{array}$

  BI (27)(7)

5. 

$\underset{0}{\overset{\infty }{\int }}\frac{1-e^{vx}}{e^x-1}dx=\psi (v)+C+\pi \cot (\pi v$