Math help

I think this is an improper integral question. Solve the integral ( going from negative infinity to infinity) cos(x)/1+x^2

Where i assume cos(x)=Re^ix

and the R just stands for real numbers.

You should replace $\cos x$ with $e^{iz}$ instead. (why? because $e^{iz} = \cos z + i \sin z$ which becomes useful at the end when you separate real/imaginary parts)

So, we have the function $f(z) = \dfrac{e^{iz}}{1+z^2}$

So then we can use contour integration, and our contour can be a semi-circle in the region $\text{Im} (z) \geq 0$, centered at the origin, with radius $R>1$. Let's call this contour $C$, which is made up of two simple arcs: a straight line on the real axis from -R to R - let's call this $\Gamma$, and the arc that goes from -R through (0,Ri) to R - let's call this arc $\gamma$.

Then we have


Can you take it from there? You can begin by applying Cauchy's integral formula, or the Residue Theorem (whichever one you have learnt), to evaluate the LHS directly.

Yes! thank you, i have ended up with pi/e as the answer by using the residue formula. lifesaver