Trying to figure out integral with infitnite limites

[z00maffect]

1. The problem statement, all variables and given/known data

\int^{\infty}_{-\infty}(1/(a^{4}+(x-x_{0})^{4}))dx

2. Relevant equations



3. The attempt at a solution

i let u = (x-x_{0})^{4}

but have no idea what to go from there

[Dick]

I would use u=(x-x0)/a and factor out the a^4. That gives you 1/(1+u^4). (1+u^4)=(u^2-sqrt(2)u+1)*(u^2+sqrt(2)u+1). Use partial fractions on that. It's not an easy integral, but it can be done.

[z00maffect]

awesome thanks! got \pi*\sqrt{2}/2

[Dick]

awesome thanks! got \pi*\sqrt{2}/2

Good job. Don't forget to put the 'a' factor back in again.

[meanyack]

As Dick said, firstly let
u=\frac{x-x_{0}}{a} then du= adx and now integral becomes


\frac{1}{a^{5}}\int^{\infty}_{-\infty}\frac{1}{1+u^{4}}du

Secondly, by letting u=e^{i\theta} and du=i*{e}^{i\theta}d\theta
you can use "residue theorem". Yet, I forgot how can we apply here. After I remember, I'll post it