integration x dx / ( 1+x^5 )?

Question

if any one could solve this i will be happy if he / she write all the steps ... :)

Answer 1

The denominator factors as

(x+1)(x² + ([sqrt(5) - 1]/2)x + 1)(x² - ([sqrt(5) + 1]/2)x + 1)

The next step is to use partial fractions. Sorry, but I'm not even going to attempt to do that here.

The answer involves inverse tangents and logs. This is what Mathematica gets:

http://i24.tinypic.com/21e9o6.gif

Since this was done by Mathematica, I have a lot of faith in its correctness (though it may not be in the simplest form possible).