Compute $\int_{|r|=2}\frac{1}{z^2+1}dz$

Question

Compute $\int_{|r|=2}\frac{1}{z^2+1}dz$. Is there a special trick to solving this? I tried letting $z(t)=2e^{it},t\in[0,2\pi]$, which gave me $$\int_{|z|=2}\frac{1}{z^2+1}dz=\int_0^{2\pi}\frac{2ie^{it}}{4e^{2it}+1}dt=\int_{0}^{2\pi}\frac{2i}{4e^{it}+e^{-it}}dt$$ but that doesn't seem to make it any easier. Is there a bettergeneral method to computing imaginary integrals along the circumference of fixed circle like this, maybe using Cauchy's formula?

Answer 1

What's problem in it... It's simple to integrate further ... Use your second Integral ... No need to divide $e^{it}$ in the numerator and denominator. Use substitution $x=e^{it}$ And integrate directly... You will get $$[\tan^{-1}(2e^{ix})]$$ Put the limits...

You will get $\boxed{0}$ as answer.