Integraal van $$$\frac{1}{\epsilon + x^{n}}$$$ met betrekking tot $$$x$$$

Bepaal $$$\int \frac{1}{\epsilon + x^{n}}\, dx$$$.

Deze integraal heeft geen gesloten vorm:

$${\color{red}{\int{\frac{1}{\epsilon + x^{n}} d x}}} = {\color{red}{\frac{x \left(1 + \frac{x^{n}}{\epsilon}\right) {{}_{2}F_{1}\left(\begin{matrix} 1, \frac{1}{n} \\ 1 + \frac{1}{n} \end{matrix}\middle| {- \frac{x^{n}}{\epsilon}} \right)}}{\epsilon + x^{n}}}}$$

Dus,

$$\int{\frac{1}{\epsilon + x^{n}} d x} = \frac{x \left(1 + \frac{x^{n}}{\epsilon}\right) {{}_{2}F_{1}\left(\begin{matrix} 1, \frac{1}{n} \\ 1 + \frac{1}{n} \end{matrix}\middle| {- \frac{x^{n}}{\epsilon}} \right)}}{\epsilon + x^{n}}$$

Vereenvoudig:

$$\int{\frac{1}{\epsilon + x^{n}} d x} = \frac{x \Phi\left(\frac{x^{n} e^{i \pi}}{\epsilon}, 1, \frac{1}{n}\right)}{\epsilon n}$$

Voeg de integratieconstante toe:

$$\int{\frac{1}{\epsilon + x^{n}} d x} = \frac{x \Phi\left(\frac{x^{n} e^{i \pi}}{\epsilon}, 1, \frac{1}{n}\right)}{\epsilon n}+C$$

$$$\int \frac{1}{\epsilon + x^{n}}\, dx = \frac{x \Phi\left(\frac{x^{n} e^{i \pi}}{\epsilon}, 1, \frac{1}{n}\right)}{\epsilon n} + C$$$A