- #1
Grothard
- 29
- 0
Homework Statement
[tex]\int_{0}^{\infty }\frac{dx}{1+x^{5}}[/tex]
The attempt at a solution
This is for my complex variables class, so I have been trying to compute it using residues. I noticed that if we extend it to the complex plane and integrate over the edges of a half disk or a quarter disk (or even 3/4 of a disk) of radius R, as R approaches infinity the circular curve part approaches zero. This means that
[tex]\int_{0}^{\infty }\frac{dx}{1+x^{5}} + \int_{0}^{\infty }\frac{dx}{1+ix^{5}} = Res(enclosed)[/tex]
and
[tex]\int_{0}^{\infty }\frac{dx}{1+x^{5}} + \int_{0}^{\infty }\frac{dx}{1-ix^{5}} = Res(enclosed)[/tex]
The residues aren't hard to compute, but I don't know what to do with the other integral
[tex]\int_{0}^{\infty }\frac{dx}{1+x^{5}}[/tex]
The attempt at a solution
This is for my complex variables class, so I have been trying to compute it using residues. I noticed that if we extend it to the complex plane and integrate over the edges of a half disk or a quarter disk (or even 3/4 of a disk) of radius R, as R approaches infinity the circular curve part approaches zero. This means that
[tex]\int_{0}^{\infty }\frac{dx}{1+x^{5}} + \int_{0}^{\infty }\frac{dx}{1+ix^{5}} = Res(enclosed)[/tex]
and
[tex]\int_{0}^{\infty }\frac{dx}{1+x^{5}} + \int_{0}^{\infty }\frac{dx}{1-ix^{5}} = Res(enclosed)[/tex]
The residues aren't hard to compute, but I don't know what to do with the other integral