I was hoping to use the Residue Theorem, but I'm not sure what contour to use; I was thinking that since $$\frac{1}{(z+1)(z^2+1)}=\frac{z-1}{z^4-1},$$ it would make sense to choose a quarter circle around the origin, as then the denominator has a simple expression on both straight-line components of the integral. Unfortunately the singularity at $z=i$ is then on the contour. Any ideas for a better contour?
$\int_0^{\infty}\frac{1}{(x+1)(x^2+1)}dx$ using complex analysis
complex-analysis
Best Answer
Too long for a comment.
It is possible to evaluate this integral via integration in the complex plane. I would do the following:
You will get $I(1-e^{2\pi ia})=2\pi i Res _{(z=-1, i, -i)}\frac{z^a}{(z+1)(z^2+1)}$ - residuals in three simple poles inside the contour.
At the end you will have to take the limit $a\to0$ (uncovering the uncertainty). All this program is not too much complicated.