一道积分题

\int_{-1}^{1}\frac{1}{\sqrt{1-x^2}(1+x^2)}dx

1st question： Anyone knows how to write equations here?


2nd question: When I first read the integral, it is in complex variables. It needs residue theorem to get the job done.

But, then I found the  path is not easy to come up with and I found someone solved it in a very strange way.

He made a substitution (1-x^2)(1+y^2)=1, or y=x/sqrt(1-x^2), then he found that this integral = tan(y)/sqrt(2).

Anyone comes up with other ideas?