## 3.6–4.1 Trigonometric Functions

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

3.611

1.

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

BI (68)(10)

2.

$\int}_{0}^{2\pi}{\left(1-cosx\right)}^{n}cosnx\text{d}x={\left(-1\right)}^{n}\frac{\pi}{{2}^{n-1}$

BI (68)(11)

3.

${\int}_{0}^{\pi}{\left(cost+isintcosx\right)}^{n}\text{d}x={\displaystyle {\int}_{0}^{\pi}{\left(cost+isintcosx\right)}^{-n-1}}\text{d}x=\pi {P}_{n}\left(cost\right)$

EH I 158(23)a

3.612

1.^{12}

$\begin{array}{lll}{\displaystyle {\int}_{0}^{\pi}\frac{sinnxcosmx}{sinx}\text{d}x}\hfill & =0\hfill & \text{for}0n\le m;\hfill \\ =\pi \hfill & \text{for}nm0,\text{if}m\text{+}n\text{isoddandpositive}\hfill \\ =0\hfill & \text{for}nm0,\text{if}m+n\text{iseven}\hfill \end{array}$

LI (64)(3)

2.^{12}

$\begin{array}{l}{\displaystyle {\int}_{0}^{\pi}\frac{sinn}{}}\hfill \end{array}$

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