Pregunta: Consider the contour shown in the following figure.Evaluate the integral\[\int_0^\infty \frac{x^a}{(x^2 + 1)^2} \, dx,\]where $-1 < a < 3$, $x^a = \exp(a \ln x)$, and show that it equals\[\frac{(1 - a) \pi}{4 \cos(a \pi / 2)}.--------------------------------------------------------------------------------------------------------------\]Considera el

Consider the contour shown in the following figure.

Evaluate the integral

$\backslash [$

$\backslash$int$\_0^\backslash$infty $\backslash$frac$\{$x$^$a$\}\{($x$^2+1)^2\}\backslash ,$ dx$,$

$\backslash ]$

where $$-1<$ a $<3$$$,$ $x$^$a $=\backslash$exp$($a $\backslash$ln x$)$$$,$ and show that it equals

$\backslash [$

$\backslash$frac$\{(1-$ a$)\backslash$pi$\}\{4\backslash$cos$($a $\backslash$pi $/2)\}.--------------------------------------------------------------------------------------------------------------$

$\backslash ]$Considera el contorno mostrado en la siguiente figura.

Figura $2$: Contorno generado por dos circunferencias de radio $R$ y $\rho ,$ donde ${\displaystyle \underset{0}{\overset{\infty }{\int }}}\frac{x^a}{(x^2+1{)}^2}dx\frac{(1-a)\pi }{4cos(a\frac{\pi }{2})}-1,y$ muestra que vale

$\frac{(1-a)\pi }{4cos(a\frac{\pi }{2})}\rho <1.$

Eval$úala$ integral

${\displaystyle \underset{0}{\overset{\infty }{\int }}}\frac{x^a}{(x^2+1{)}^2}dx$

donde $-1,y$ muestra que vale

$\frac{(1-a)\pi }{4cos(a\frac{\pi }{2})}$