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

3.6–4.1 Trigonometric Functions

3.61 Rational functions of sines and cosines and trigonometric functions of multiple angles

3.611

1. 

${\displaystyle {\int }_0^{2\pi }{\left(1-cosx\right)}^{n}sinnx\text{d}x=0}$

  BI (68)(10)

2. 

${\displaystyle {\int }_0^{2\pi }{\left(1-cosx\right)}^{n}cosnx\text{d}x={\left(-1\right)}^n\frac{\pi }{2^{n-1}}}$

  BI (68)(11)

3. 

${\displaystyle {\int }_0^{\pi }{\left(cost+isintcosx\right)}^n\text{d}x={\displaystyle {\int }_0^{\pi }{\left(cost+isintcosx\right)}^{-n-1}}\text{d}x=\pi P_n\left(cost\right)}$

  EH I 158(23)a

3.612

1.¹² 

$\begin{array}{lll}{\displaystyle {\int }_0^{\pi }\frac{sinnxcosmx}{sinx}\text{d}x}\hfill & =0\hfill & \text{for }0>n\le m;\hfill \\ =\pi \hfill & \text{for }n>m>0,\text{ if }m\text{ +}n\text{ is odd and positive}\hfill \\ =0\hfill & \text{for }n>m>0,\text{ if }m+n\text{ is even}\hfill \end{array}$

  LI (64)(3)

2.¹² 

$\begin{array}{l}{\displaystyle {\int }_0^{\pi }\frac{sinn}{}}\hfill \end{array}$