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

2.63–2.65 Products of trigonometric functions and powers

2.631

1. 

$\begin{array}{ccc}{\displaystyle \int x^r{\sin }^{p}x{\text{cos}}^{q}x\text{d}x}\hfill & =\hfill & \frac{1}{{\left(p\text{+}q\right)}^2}[\left(p\text{+}q\right)x^r{\sin }^{p+1}x{\text{cos}}^{q-1}x\hfill \\ +r{x}^{r-1}{\sin }^{p}x{\text{cos}}^{q}x-r\left(r-1\right){\displaystyle \int x^{r-2}{\sin }^{p}x{\text{cos}}^{q}x\text{d}x}\hfill \\ -rp{\displaystyle \int x^{r-1}{\sin }^{p-1}x{\text{cos}}^{q–1}x\text{d}x+\left(q-1\right)\left(p+q\right){\displaystyle \int x^r{\sin }^{p}x{\text{cos}}^{q–2}x\text{d}x}}]\hfill \\ =\hfill & \frac{1}{{\left(p\text{+}q\right)}^2}[-\left(p\text{+}q\right)x^r{\sin }^{p-1}x{\text{cos}}^{q+1}x\hfill \\ +r{x}^{r-1}{\sin }^{p}x{\text{cos}}^{q}x-r\left(r-1\right){\displaystyle \int x^{r-2}{\sin }^{p}x{\text{cos}}^{q}x\text{d}x}\hfill \\ +rp{\displaystyle \int x^{r-1}{\sin }^{p-1}x{\text{cos}}^{q-1}x\text{d}x+\left(p-1\right)\left(p+q\right){\displaystyle \int x^r{\sin }^{p-2}x{\text{cos}}^{q}x\text{d}x}}]\hfill \end{array}$

GU (331)(1)

2. 

$\begin{array}{ccc}{\displaystyle \int x^m{\sin }^{n}x\text{d}x}\hfill & =\hfill & \frac{x^{m-1}{\sin }^{n-1}x}{n^2}\{m\text{sin}x-nx\text{cos ...}\hfill \end{array}$