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

$\begin{array}{l}{\displaystyle \int \frac{x^n}{\text{sin }x}}\text{d}x\text{ =}\frac{x^n}{n}+{\displaystyle \sum _{k=1}^{\infty }{\left(-1\right)}^{k+1}\frac{2\left(2^{2k-1}-1\right)}{\left(n+2k\right)\left(2k\right)!}{B}_{2k}{x}^{n+2k}}\hfill \\ \left[\left|x\right|<\pi ,n>0\right]\hfill \end{array}$

TU (333)(8b)

4.¹² 

$\begin{array}{l}{\displaystyle \int \frac{x^n}{x^n\text{sin}x}}\text{d}x=-\frac{1}{n{x}^n}-\left[1+{\left(-1\right)}^n\right]{\left(-1\right)}^{\frac{n}{2}}\frac{2^{n-1}-1}{n!}{B}_{n}lnx-{\displaystyle \sum _{\begin{array}{l}k=1\hfill \\ k\ne \frac{n}{2}\hfill \end{array}}^{\infty }{\left(-1\right)}^k\frac{2\left(2^{2n}-1\right)}{\left(2k-n\right)\cdot \left(2k\right)!}{B}_{2k}{x}^{2k-n}}\hfill \\ \left[n>1,\left|x\right|<\pi ,\right]\hfill \end{array}$

GU (333)(9b)

5.⁸ 

${\displaystyle \int \frac{x^n\text{d}x}{\text{cos}x}={\displaystyle \sum _{k=0}^{\infty }\frac{\left|E_{2k}\right|x^{n+2k+1}}{\left(n+2k+1\right)\left(2k\right)!}}}\left[\left|x\right|<\frac{\pi }{2},n>0\right]$

GU (333)(10b)

6. 

$\begin{array}{l}{\displaystyle \int \frac{\text{d}x}{x^n\text{cos}x}=\frac{1}{2}\left[1-{\left(-1\right)}^n\right]}\hfill \end{array}$