calculate-A-n-0-dx-n-x-n-2-with-n-gt-1-

Question Number 66467 by mathmax by abdo last updated on 15/Aug/19

c a l c u l a t e A_{n} = \int_{0}^{\infty} \frac{d x}{{(n + x^{n})}^{2}} w i t h n > 1

Commented by mathmax by abdo last updated on 16/Aug/19

w e h a v e p r o v e d \int_{0}^{\infty} \frac{d x}{{(a + x^{n})}^{2}} = \frac{π}{n s i n (\frac{π}{n})} (1 - \frac{1}{n}) a^{\frac{1}{n} - 2}

a = n \Rightarrow \int_{0}^{\infty} \frac{d x}{{(n + x^{n})}^{n}} = \frac{π}{n s i n (\frac{π}{n})} (1 - \frac{1}{n}) n^{\frac{1}{n} - 2}

= \frac{π (n - 1)}{n^{2} s i n (\frac{π}{n})} (^{n} \sqrt{n}) \times \frac{1}{n^{2}} = \frac{π (n - 1) (^{n} \sqrt{n})}{n^{4} s i n (\frac{π}{n})} w i t h n > 1