Complex Integration Using Residue Theorem with branch cut

Question

How to calculate $$\int_{0}^{\infty }\frac{ln(1+x^{2})}{(1+x^{2})} dx $$ ? How to choose the branch cut and proceed?

Answer 1

First, as $\frac{ln(1+x^{2})}{1+x^{2}}$ is an even function, $\int_{0}^{\infty }\frac{ln(1+x^{2})}{(1+x^{2})} dx = \frac{1}{2} \int_{- \infty}^{\infty }\frac{ln(1+x^{2})}{(1+x^{2})} dx$.

Now, there's a general theorem which says that if $lim_{z \to \infty}zf(z)=0$ then \begin{equation} \int_{- \infty}^{\infty }f(x) dx = 2 \pi i \sum_{\substack{a \in Sing(f) \\ a \in \mathbb{H}}}Res(f,a) \ + \ \pi i \sum_{\substack{a \in Sing(f) \\ Im(a) = 0}}Res(f,a) \end{equation} where $\mathbb{H}= \{\,z \in \mathbb{C}: Im(z) > 0\,\} $.

You know that $i$ and $-i$ are the only singularities of your function. As $-i \not\in \mathbb{H}$ your problem is reduced to find $Res(f,i)$.