I am trying to learn how to work with branch points. I decided to change the sign in the next classical example:
$$f(x)=\int_{-1}^{+1} \frac{1}{\sqrt{1-x^2}}$$
In this example, simply select the path of integration by drawing a cut along the active axis from -1 to +1. Answer is known - it's $\pi$. I changed the sign of the radical expression, and now I'm trying to solve an integral of the next form:
$$f(x)=\int_{-1}^{+1} \frac{1}{\sqrt{1+x^2}}$$
And here I ran into difficulties - I don’t know which path of integration I should choose, since the branch points are on the imaginary axis, and I need to reduce the integral along the closed path to the integral along the real axis from -1 to 1.
If we do by analogy with the original number, then I come to the fact that the integral now varies from -i to i and does not reduce to the integral of a real variable.
Could you please explain which contour should be chosen?