MHB What is the method for integrating 1/(1+x^n) using roots of polynomials?

[juantheron]

$(1)\displaystyle \;\; \int\frac{1}{1+x^6}dx$

$(2)\displaystyle \;\; \int\frac{1}{1+x^8}dx$

 

[chisigma]

$(1)\displaystyle \;\; \int\frac{1}{1+x^6}dx$

$(2)\displaystyle \;\; \int\frac{1}{1+x^8}dx$

May be is useful to consider the general case...

$\displaystyle \int \frac{dx}{1+x^{n}}$ (1)

The root of the polinomial $\displaystyle 1+x^{n}$ are $\displaystyle x_{k}= e^{i\ \frac{2k+1}{n}\ \pi}\ ,\ k= 0.1,...,n-1$ so that is...

$\displaystyle \frac{1}{1+x^{n}} = \sum_{k=0}^{n-1} \frac{r_{k}}{x-x_{k}}$ (2)

... where...

$\displaystyle r_{k}= \lim_{x \rightarrow x_{k}} \frac{x-x_{k}}{1+x^{n}}$ (3)

Now You can integrate (2) 'term by term' obtaining... $\displaystyle \int \frac{dx}{1+x^{n}}= \sum_{k=0}^{n-1} r_{k}\ \ln (x-x_{k}) + c $ (4)

Kind regards

$\chi$ $\sigma$