Sorry for my English if there is any mistake. The exercice for which I need help is the following:
Compute using complex methods: $I=\int_1 ^\infty \frac{\mathrm{d}x}{x^2+1}$
i) Choose the complex function to integrate.
I guess it is $f(z)=1/(z^2+1)$
ii) Choose the contour.
I don't know what to do here. In my notes there are only examples when the integral is from $-\infty$ to $\infty$, so it takes a circumference of radius $r$ and lets it tend to $\infty$.
iii) Compute the integrals along circumferences.
iv) Compute the branch cut.
I don't know why is this question here, because the function is not multivalued.
v) Compute the integral.
vi) Compute the integral using elemental methods.
$I=\int_1 ^\infty \frac{\mathrm{d}x}{x^2+1}=\lim _{a\rightarrow \infty} \int _1 ^a \frac{\mathrm{d}x}{x^2+1}= \lim _{a\rightarrow \infty} \left[ \arctan x \right]_1 ^a =\frac{\pi}{2}-\frac{\pi}{4}=\frac{\pi}{4}$
Edit: The answer might follow the steps given. My teacher did an exercice that way, but I don't know why he uses such method (the example is in a comment within the answers).