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

$\begin{array}{ll}{\displaystyle {\int }_0^{\infty }\frac{\text{d}x}{x^{n+1}}{\displaystyle \prod _{k=0}^n\text{sin}\left(a_{k}x\right)=\frac{\pi }{2}}{\displaystyle \prod _{k=1}^n{a}_k}}\hfill & \left[\begin{array}{ll}{a}_0>{\displaystyle \sum _{k=1}^n{a}_k,}\hfill & a_k>0\hfill \end{array}\right]\hfill \end{array}$

  FI II 646

2. 

$\begin{array}{ll}{\displaystyle {\int }_0^{\infty }\frac{sin\left(ax\right)}{x^{n+1}}\text{d}x{\displaystyle \prod _{k=1}^n\text{sin}\left(a_{k}x\right){\displaystyle \prod _{j=1}^{m}cos\left(b_{j}x\right)=\frac{\pi }{2}}}{\displaystyle \prod _{k=1}^n{a}_k}}\hfill & \left[a>{\displaystyle \sum _{k=1}^n\left|a_k\right|+{\displaystyle \sum _{j=1}^m\left|b_j\right|}}\right]\hfill \end{array}$

  WH

3.747

1.⁷ 

${\displaystyle {\int }_0^{\pi /2}\frac{x^m}{\text{sin}x}\text{d}x={\left(\frac{\pi }{2}\right)}^m\left[\frac{1}{m}+{\displaystyle \sum _{k=1}^{\infty }\frac{{\text{2}}^{\text{2}k-\text{1}}-\text{1}}{4^{2k-1}(m+\text{2}k)}\zeta (\text{2}k)}\right]=\text{2}\pi G-\frac{7}{2}\zeta (\text{3})}$

  LI (206)(2)

$\left[m=2\right]$

2. 

${\displaystyle {\int }_0^{\pi /2}\frac{x\text{d}x}{\text{sin}x}}={\displaystyle {\int }_0^{\pi /2}\frac{\left(\frac{\pi }{2}-x\right)\text{d}x}{cosx}=2G}$

  BI(204)(18), BI(206)(1), GW(333)(32)

3.¹² 

$\begin{array}{l}{\displaystyle {\int }_0^{\infty }\frac{x}{(x^{\text{2}}+a^{\text{2}})\text{sin}x}\text{d}x=}\hfill \end{array}$