Solving Integrals with Contour Integrals and Cauchy PV

[pleasehelpmeno]

In trying to solve \int^{\infty}_{-\infty} x + \frac{1}{x} dx could it be split up and solved using the Cauchy Principle Value theorem and a contour integral along a semi-circle. Thus;
PV\int^{\infty}_{-\infty}x dx =0 +\int \frac{1}{x} dx = \int^{\pi}_{0} i d\theta

Is this valid reasoning?

 

[HallsofIvy]

No, it isn't. For one thing, you have a pole at x= 0 which is on your contour. You would need another semicircle around x= 0 to avoid that.

 

[pleasehelpmeno]

The initial intergal was f(x)=\int^{\infty}_{-\infty} \sqrt{x^{2}+y^{2}}dx so I taylor expanded it to get f(x) \approx \int^{\infty}_{-\infty} x + \frac{y^{2}}{2x} dx

I thought one could then justify that the cauchy principle value of \int^{\infty}_{-\infty} x dx =0 and then what I have done with the \frac{1}{x} integral. I am doubting my approach because the Taylor series was about x=0 which seems odd, is there a better way? I read that one take x to be complex then contour interagtes it, I am just not sure how?