# Trying to figure out integral with infitnite limites

## Homework Statement

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

## The Attempt at a Solution

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

but have no idea what to go from there

Last edited:

## Answers and Replies

Dick
Science Advisor
Homework Helper
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.

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

Dick
Science Advisor
Homework Helper
awesome thanks! got $$\pi*\sqrt{2}/2$$

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

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