How to Calculate PV$\int_0^{\infty}\frac{\cos ax}{x^4-1}\:dx\: $for all a in R

Question

Can anyone help calculate this integral?

$$\text{PV}\int_0^{\infty}\frac{\cos ax}{x^4-1}\,dx$$

for all $a \in \mathbb{R}$.

Thanks in advance!

Answer 1

To help you begin, first note that

$$\frac{1}{x^4-1}=\frac{1/2}{x^2-1}-\frac{1/2}{x^2+1}$$

Hence, we have

$$\begin{align} \text{PV}\int_0^\infty \frac{\cos(ax)}{x^4-1}\,dx&=\frac12\text{PV}\int_{-\infty}^\infty \frac{\cos(ax)}{x^4-1}\,dx\\\\ &=\frac12 \text{PV}\int_{-\infty}^\infty \frac{\cos(ax)}{x^2-1}\,dx-\frac12\int_{-\infty}^\infty \frac{\cos(ax)}{x^2+1}\,dx \end{align}$$

To finish, write $\cos(ax)=\text{Re}(e^{i|a|x})$, and close the real line contours in the upper half-plane, deform the contour around the singularities around $z=\pm 1$ of the first integral on the right-hand side, and apply the reside theorem to the second integral to account for the poles at $z=\pm i$.

Can you finish now?

Answer 2

Hint: Here is a contour that might prove useful