How to integrate this integral using contour integration?
$$\int_1^{\infty}\frac{\sqrt{x-1}}{(1+x)^2}dx$$
2026-04-01 08:02:53.1775030573
Integrate $\int_1^{\infty}\frac{\sqrt{x-1}}{(1+x)^2}dx$ using contour integration
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Hint: Take a disk of radius $R$ centered at $1$ and remove an $r$-neighborhood of the segment $[1,R+1]$. The boundary of this consists of an almost complete circle large circle, half a small circle (radius $r$) and two segments (above and below the real segment $[1,R+1]$ and your function takes opposite signs on the two because of the square root. The contribution of a contour integral along the segments give you therefore twice the value you want and you pick up a residue at $x=-1$ as you let $R\rightarrow \infty$ and $r\rightarrow 0$.