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

6-7

Definite Integrals of Special Functions

6.1 Elliptic Integrals and Functions

Notation: $k^{\prime }=\sqrt{1-k^2}$(cf.8.1).

6.11 Forms containing F (x, k)

6.111

${\displaystyle {\int }_0^{\pi /2}F\left(x,k\right)cotx\text{d}x=\frac{\pi }{4}\mathit{K}}\left(k^{\prime }\right)+\frac{1}{2}lnk\mathit{K}\left(k\right)$

  BI (350)(1)

6.112

1. 

${\displaystyle {\int }_0^{\pi /2}F(x,k)}\frac{sinxcosx}{1+k{sin}^{2}x}\text{d}x=\frac{1}{4k}\mathit{K}(k)ln\frac{(1+k)\sqrt{k}}{2}+\frac{\pi }{16k}\mathit{K}(k^{\prime })$

  BI (350)(6)

2. 

${\displaystyle {\int }_0^{\pi /2}F(x,k)}\frac{sinxcosx}{1-k{sin}^{2}x}\text{d}x=\frac{1}{4k}\mathit{K}(k)ln\frac{2}{(1-k)\sqrt{k}}-\frac{\pi }{16k}\mathit{K}(k^{\prime })$

  BI (350)(7) ...