calculate A_n =∫_(−∞) ^(+∞) (dx/((x^2 +1)(x^2 +2)....(x^2 +n))) with n integr natural and n≥1


Question and Answers Forum

All Questions Topic List
Integration Questions
Previous in All Question Next in All Question
Previous in Integration Next in Integration
Question Number 64166 by mathmax by abdo last updated on 14/Jul/19

$c a l c u l a t e A_{n} = \int_{- \infty}^{+ \infty} \frac{d x}{(x^{2} + 1) (x^{2} + 2) . . . . (x^{2} + n)}$ $w i t h n i n t e g r n a t u r a l a n d n ⩾ 1$
	Answered by Eminem last updated on 14/Jul/19

	$f (z) = \frac{d z}{(z^{2} + 1) . . . . . . (z^{2} + n)}$ $A_{n} = 2 i π R e (z k = i \sqrt{k}, k \in [1 . . n])$ $r e s (f . i \sqrt{k}) = \frac{1}{2 i \sqrt{k} \underset{j = 1}{\prod^{n, k \neq j}} (1 - j)} = \frac{1 - k}{2 i \sqrt{k} {(- 1)}^{n} (n - 1)!}$ $A_{n} = 2 i π \underset{k = 2}{\sum^{n}} \frac{(1 - k)}{2 i \sqrt{k} (n - 1)!} + \frac{2 i π}{2 i {(- 1)}^{n} (n - 1)!}$
Terms of Service
Privacy Policy
Contact: info@tinkutara.com

Question and Answers Forum