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

2.14 Forms containing the binomial $1\pm x^n$

2.141

1. 

${\displaystyle \int \frac{\text{d}x}{1+x}=ln\left(\text{1+}x\right)}$

2.¹¹ 

${\displaystyle \int \frac{\text{d}x}{1\text{+}{x}^2}=\text{arctan}x=-\text{arctan}\left(\frac{1}{x}\right)}$

(see also2.124 1)

3. 

${\displaystyle \int \frac{\text{d}x}{1\text{+}{x}^3}=\frac{1}{3}ln\frac{1+x}{\sqrt{1-x+x^2}}\text{+}\frac{1}{\sqrt{3}}\text{arctan}\frac{x\sqrt{3}}{2-x}}$

(see also2.126 1)

4. 

${\displaystyle \int \frac{\text{d}x}{1\text{+}{x}^4}=\frac{1}{4\sqrt{2}}ln\frac{1+x\sqrt{2}+x^2}{1-x\sqrt{2}+x^2}\text{+}\frac{1}{2\sqrt{2}}\text{arctan}\frac{x\sqrt{2}}{1-x^2}}$

(see also2.132 1)

2.142

$\begin{array}{l}\begin{array}{ccc}{\displaystyle \int \frac{\text{d}x}{1\text{+}{x}^{\text{n}}}}& =& -\frac{2}{n}{\displaystyle \sum _{k=0}^{\frac{n}{2}-1}{P}_k\text{cos}}\left(\frac{2k+1}{n}\pi \right)+\frac{2}{n}{\displaystyle \sum _{k=0}^{\frac{n}{2}-1}{Q}_k\text{sin ...}}\end{array}\end{array}$