Daniel Ward

Posted on Thursday, 08 May, 2003 - 08:03 pm:

Given that

I (n) = \int_{0}^{1} (x^{2} + a^{2})^{- n}

show that

(2 {na}^{2}) I (n + 1) = (2 n - 1) I (n) + (1 + a^{2})^{- n} .

Splitting up to do it by parts leaves you with a nasty arctan wandering around loose.

Help!

Oscar Randal-Williams

Posted on Friday, 09 May, 2003 - 10:35 am:

If you write LaTeX Image

then

.

If you then use parts on the first integral it works fine.

David Loeffler

Posted on Friday, 09 May, 2003 - 03:58 pm:

When doing the integral by parts, it might help to split it up as

(x) ( x / (x² + a² )ⁿ⁺¹ ), as it is obvious how to integrate the second factor. Using (x² ) ( 1 / (x² + a² )ⁿ⁺¹ ) will give you the nasty arctans.