So, I know how to find the area between curves when they are bounded, but what do I do here when they are unbounded. I have learned about improper integrals, but I am not sure how I can use them here. I have already done some work, but I'm not sure if it is correct or if I am going in the right direction.
Can anyone suggest how I can start?
It is the integral
$$\int_2^\infty\left(\frac2{x-1}-\frac{2x}{x^2+1}\right)\,dx=\left.\lim_{b\to\infty}\left[2\log(x-1)-\log(x^2+1)\right]\right|_2^b=$$
$$\left.\lim_{b\to\infty}\log\frac{(x-1)^2}{x^2+1}\right|_2^b=\lim_{b\to\infty}\left[\log\frac{(b-1)^2}{b^2+1}-\log\frac15\right]=\log1-\log\frac15=\log5$$