dx-1-x-10-

Question Number 164171 by mkam last updated on 15/Jan/22

\int \frac{d x}{1 + x^{10}}

Answered by Ar Brandon last updated on 15/Jan/22

\int \frac{d x}{1 + x^{10}} = \int \underset{n = 0}{\sum^{\infty}} {(- x^{10})}^{n} d x = \sum_{n ⩾ 0} \frac{{(1)}_{n} {(- 1)}^{n} x^{10 n + 1}}{n! (10 n + 1)}

= \frac{x}{10} \sum_{n ⩾ 0} \frac{{(1)}_{n} {(- x^{10})}^{n}}{n! (n + \frac{1}{10})} = \frac{x}{10} \sum_{n ⩾ 0} \frac{{(1)}_{n} Γ (n + \frac{1}{10})}{n! Γ (n + \frac{11}{10})} {(- x^{10})}^{n}

= \frac{x}{100} \sum_{n ⩾ 0} \frac{{(1)}_{n} {(\frac{1}{10})}_{n}}{n! {(\frac{11}{10})}_{n}} {(- x^{10})}^{n} = \frac{x}{100} \underset{2}{} F_{1} (\frac{1}{10}, 1; \frac{11}{10}; - x^{10}) + C