### 3.64–3.65 Powers and rational functions of trigonometric functions

3.641

1.

${\int }_{0}^{\pi /2}\frac{{sin}^{p-1}x{cos}^{-p}x}{acosx+bsinx}\text{d}x={\int }_{0}^{\pi /2}\frac{{sin}^{-p}x{cos}^{p-1}x}{asinx+bcosx}\text{d}x=\frac{\pi cos\text{​}\text{?}\text{ec}p\pi }{{a}^{1-pbp}}$

GW (331)(62)

[ab > 0, 0 < p < 1]

2.

${\int }_{0}^{\pi /2}\frac{{sin}^{1-p}x{cos}^{p}x}{{{{{{{{\left(sinx+cosx\right)}^{3}}^{}}^{}}^{}}^{}}^{}}^{}}\text{?}\text{d}x={\int }_{0}^{\pi /2}\frac{{sin}^{p}x{cos}^{1-p}x}{{\left(sinx+cosx\right)}^{3}}\text{d}x=\frac{\left(1-p\right)p}{2}\pi cos\text{ec}p\pi$

BI(48)(5)

[-1 < p < 2]

3.642

1.

BI (48)(28)

2.

$\begin{array}{l}{\int }_{0}^{\pi /2}\frac{{sin}^{n-1}x{cos}^{n-1}x\text{d}x}{{{{{{{{\left({a}^{2}{cos}^{2}x+{b}^{2}{sin}^{2}x\right)}^{}}^{}}^{}}^{}}^{}}^{}}^{}}\hfill \end{array}$

Get Table of Integrals, Series, and Products, 8th Edition now with O’Reilly online learning.

O’Reilly members experience live online training, plus books, videos, and digital content from 200+ publishers.