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

FI II 646

2.

$\begin{array}{ll}{\int }_{0}^{\infty }\frac{sin\left(ax\right)}{{x}^{n+1}}\text{d}x\prod _{k=1}^{n}\text{sin}\left({a}_{k}x\right)\prod _{j=1}^{m}cos\left({b}_{j}x\right)=\frac{\pi }{2}\prod _{k=1}^{n}{a}_{k}\hfill & \left[a>\sum _{k=1}^{n}|{a}_{k}|+\sum _{j=1}^{m}|{b}_{j}|\right]\hfill \end{array}$

WH

3.747

1.7

${\int }_{0}^{\pi /2}\frac{{x}^{m}}{\text{sin}x}\text{d}x={\left(\frac{\pi }{2}\right)}^{m}\left[\frac{1}{m}+\sum _{k=1}^{\infty }\frac{{\text{2}}^{\text{2}k-\text{1}}-\text{1}}{{4}^{2k-1}\left(m+\text{2}k\right)}\zeta \left(\text{2}k\right)\right]=\text{2}\pi G-\frac{7}{2}\zeta \left(\text{3}\right)$

LI (206)(2)

$[m=2]$

2.

${\int }_{0}^{\pi /2}\frac{x\text{d}x}{\text{sin}x}={\int }_{0}^{\pi /2}\frac{\left(\frac{\pi }{2}-x\right)\text{d}x}{cosx}=2G$

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

3.12

$\begin{array}{l}{\int }_{0}^{\infty }\frac{x}{\left({x}^{\text{2}}+{a}^{\text{2}}\right)\text{sin}x}\text{d}x=\hfill \end{array}$

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