# 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

${\int}_{0}^{\pi /2}F\left(x,k\right)cotx\text{d}x=\frac{\pi}{4}\mathit{K}}\left({k}^{\prime}\right)+\frac{1}{2}lnk\mathit{K}\left(k\right)$

BI (350)(1)

6.112

1.

${\int}_{0}^{\pi /2}F(x,k)}\frac{sinxcosx}{1+k{sin}^{2}x}\text{d}x=\frac{1}{4k}\mathit{K}(k)ln\frac{(1+k)\sqrt{k}}{2}+\frac{\pi}{16k}\mathit{K}({k}^{\prime})$

BI (350)(6)

2.

${\int}_{0}^{\pi /2}F(x,k)}\frac{sinxcosx}{1-k{sin}^{2}x}\text{d}x=\frac{1}{4k}\mathit{K}(k)ln\frac{2}{(1-k)\sqrt{k}}-\frac{\pi}{16k}\mathit{K}({k}^{\prime})$

BI (350)(7) ...

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.