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### 3.82–3.83 Powers of trigonometric functions combined with other powers

3.821

1.

${\int }_{0}^{\pi }x{sin}^{p}x\text{d}x=\frac{{\pi }^{2}}{{2}^{p+1}}\frac{\Gamma \left(p+1\right)}{{\left[\Gamma \left(\frac{p}{2}+1\right)\right]}^{2}}$

BI(218)(7), LO V 121(71)

[p > −1]

2.

$\begin{array}{lll}{\int }_{0}^{r\pi }x{sin}^{n}x\text{d}x\hfill & =\frac{{\pi }^{2}}{2}\frac{\left(2m-1\right)!!}{\left(2m\right)!!}{r}^{2}\hfill & \left[n=2m\right]\hfill \\ ={\left(-1\right)}^{r+1}\pi \frac{\left(2m\right)!!}{\left(2m+1\right)!!}r\hfill & \left[n=2m+1\right]\hfill \end{array}$

GW (333)(8c)

[r is a natural number]

3.11

GW (333)(9b)

4.

${\int }_{0}^{\pi }x{cos}^{2m}x\text{d}x=\frac{{\pi }^{2}}{2}\frac{\left(2m-1\right)!!}{\left(2m\right)!!}$

BI (218)(10) ...

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