Integral compute

yzymat

compute the Integral

**Posted:** Sun Nov 08, 2009 2:42 am

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AndrewTom

How about partial fractions with denominators $1-x, 1+x, 1+x^{2}, 1 +\sqrt{2}x+x^{2}, 1-\sqrt{2}x+x^{2}$ , giving logs and arctans?

**Posted:** Sun Nov 08, 2009 6:00 am

*210* · Hodge Conjecture Offline Joined: 07 Jul 2008 Posts: 52

Hello !

hint

$f(x)= \frac{4\sqrt{2}}{x^4 +1} + \frac{-8x^3}{(1-x^2)(1+x^4)}$

$x^4+1=( 1 +\sqrt{2}x+x^{2})( 1-\sqrt{2}x+x^{2})$

**Posted:** Sun Nov 08, 2009 6:12 am

akech

Note that:
$\frac{4\sqrt{2} - 8x^3 - 4\sqrt{2}x^4 - 8x^5}{1 - x^8} = \frac{4\sqrt{2}\left(1 - x^4\right)}{(1 - x^4)(1 + x^4)} - \frac{8x^...$

**Posted:** Sun Nov 08, 2009 6:18 am

Dr Sonnhard Graubner

hello, i think the result of your integral is $\pi$ .
Sonnhard.

**Posted:** Sun Nov 08, 2009 6:31 am

yzymat

Yes . the result is $\pi$
thanks

**Posted:** Sun Nov 08, 2009 7:07 am

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