How can I solve the integral of 1/(1+x^4) from 0 to ∞?

[vokurka]

∫1/(1+x^4) dx, from 0 to ∞


I have tried integration per partes, several different substitutions and transformation into different coordinate system but i have always only found another equivalent integral that i was not able to solve... I have also performed a numerical integration, but i need an analytical solution...
Thanks

 

[SteamKing]

You should be able to work out the indefinite integral with one substitution. By the way, what is the derivative of the arc tangent?

 

[Ray Vickson]

∫1/(1+x^4) dx, from 0 to ∞


I have tried integration per partes, several different substitutions and transformation into different coordinate system but i have always only found another equivalent integral that i was not able to solve... I have also performed a numerical integration, but i need an analytical solution...
Thanks

Partial fractions.

 

[Curious3141]

∫1/(1+x^4) dx, from 0 to ∞ I have tried integration per partes, several different substitutions and transformation into different coordinate system but i have always only found another equivalent integral that i was not able to solve... I have also performed a numerical integration, but i need an analytical solution...
Thanks

Hint: ##x^4 + 1 = (x^2+1)^2 - (\sqrt{2}x)^2##. Rearrange, factorise.

Then use what Ray suggested.

 

[SithsNGiggles]

Try substituting ##\sqrt{u}=x##.

Edit: Sorry, this doesn't actually work. Disregard!

 

[Dick]

You have a definite integral. I'd suggest using contour integration and the residue theorem if you know that technique.