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

$\begin{array}{ll}{\displaystyle {\int }_0^{\infty }\frac{x^{4n+3}{e}^{-px}}{a^{\text{4}}-x^{\text{4}}}\text{d}x}\hfill & =\frac{\text{1}}{\text{4}}{a}^{\text{4}n}\left[e^{ap}\text{Ei}(-ap)+e^{-ap}\text{Ei}(ap)+\text{2}\text{ci}(ap)cosap+\text{2}\text{si}(ap)sinap\right]\hfill \\ -\frac{\text{1}}{p^{\text{4}n}}{\displaystyle \sum _{k=1}^n(\text{4}n-\text{4}k+3)!{\left(a^{\text{4}}{p}^{\text{4}}\right)}^{k-\text{1}}}\hfill \end{array}$

  BI (91) (25)

$\left[p>0,a>0\right]$

3.359

$\begin{array}{ll}{\displaystyle {\int }_{-\infty }^{\infty }\frac{{(i-x)}^n{e}^{-ipx}}{{(i+x)}^{n}i+x^{\text{2}}}\text{d}x}\hfill & ={(-\text{1})}^{n-1}\text{2}\pi p{e}^{-p}{L}_{n-\text{1}}(\text{2}p)\hfill \\ =0\hfill \end{array}\begin{array}{l}\text{for}p>0;\hfill \\ \text{for}p<0.\hfill \end{array}$

  ET I 118(2)

3.36–3.37 Combinations of exponentials and algebraic functions

3.361

1.⁸ 

${\displaystyle {\int }_0^u\frac{e^{-qx}}{\sqrt{x}}\text{d}x=\sqrt{\frac{\pi }{q}}\Phi \left(\sqrt{qu}\right)}[q>0]$

2.⁸ 

${\displaystyle {\int }_0^{\infty }\frac{e^{-qx}}{\sqrt{x}}\text{d}x=\sqrt{\frac{\pi }{q}}}[q>0]$