Complex Integral: Struggling to Integrate [(lnx)^2](1+x^2)

[sachi]

We have to integrate [(lnx)^2)/(1+(x^2)) from zero to infinity.
I have set up the correct complex integral with a branch cut along the negative y-axis, but I end up with an integral of [(lnx)^2](1+x^2) from minus infinity to zero. I'm not sure how to deal with this.

 

[shmoe]

We have to integrate [(lnx)^2)/(1+(x^2)) from zero to infinity.
I have set up the correct complex integral with a branch cut along the negative y-axis, but I end up with an integral of [(lnx)^2](1+x^2) from minus infinity to zero. I'm not sure how to deal with this.

On the negative real axis replace ln(x) by ln|x|+Pi*i. Expand the square and you'll have 3 integrals, one is the same as the thing you're after, one can be evaluated using first year calculus stuff, and one is necessarily zero (compare imaginary parts)