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

3.64–3.65 Powers and rational functions of trigonometric functions

3.641

1. 

${\displaystyle {\int }_0^{\pi /2}\frac{{sin}^{p-1}x{cos}^{-p}x}{acosx+bsinx}}\text{d}x={\displaystyle {\int }_0^{\pi /2}\frac{{sin}^{-p}x{cos}^{p-1}x}{asinx+bcosx}\text{d}x=\frac{\pi cos\text{}\text{?}\text{ec}p\pi }{a^{1-pbp}}}$

  GW (331)(62)

[ab > 0, 0 < p < 1]

2. 

${\displaystyle {\int }_0^{\pi /2}\frac{{sin}^{1-p}x{cos}^{p}x}{{{{{{{{\left(sinx+cosx\right)}^3}^{}}^{}}^{}}^{}}^{}}^{}}\text{?}}\text{d}x={\displaystyle {\int }_0^{\pi /2}\frac{{sin}^{p}x{cos}^{1-p}x}{{\left(sinx+cosx\right)}^3}\text{d}x=\frac{\left(1-p\right)p}{2}\pi cos\text{ec}p\pi }$

  BI(48)(5)

[-1 < p < 2]

3.642

1. 

$\begin{array}{ll}{\displaystyle {\int }_0^{\pi /2}\frac{{sin}^{2\mu -1}x{cos}^{2\nu -1}x\text{d}x}{{{{{{{{\left(a^2{sin}^{2}x+b^2{cos}^{2}x\right)}^{\mu +\nu }}^{}}^{}}^{}}^{}}^{}}^{}}\text{?}}=\frac{1}{2a^{2\mu }{b}^{2\nu }}\text{B}\left(\mu ,\nu \right)\hfill & \left[\mathrm{Re}\mu >0,\mathrm{Re}\nu >0\right]\hfill \end{array}$

  BI (48)(28)

2. 

$\begin{array}{l}{\displaystyle {\int }_0^{\pi /2}\frac{{sin}^{n-1}x{cos}^{n-1}x\text{d}x}{{{{{{{{\left(a^2{cos}^{2}x+b^2{sin}^{2}x\right)}^{}}^{}}^{}}^{}}^{}}^{}}^{}}}\hfill \end{array}$