To calculate the definite integral $I = \int_{0}^{\infty} \frac{x^{\beta}}{(x^{2}+1)^{2}} \mathrm{~d}x$, consider the line integral of the complex function $f(z) = \frac{z^{\beta}}{(z^{2}+1)^{2}}$ on the complex plane. Here, $\beta$ is a real number and $0 < \beta < 1$. The closed integration path $C = C_{1} + C_{R} + C_{2} + C_{r}$ $(0 < r < 1 < R)$ is defined with semicircles and line segments as shown in Figure 3.1.
Using the residue theorem, calculate the line integral $\oint_{C} f(z) \mathrm{d}z$.
Express $\int_{C_{1}} f(z) \mathrm{d}z + \int_{C_{2}} f(z) \mathrm{d}z$ with the definite integral $\int_{r}^{R} \frac{x^{\beta}}{(x^{2}+1)^{2}} \mathrm{~d}x$.
Using the previous results, calculate the definite integral $I$.
To calculate the definite integral $I = \int_{0}^{\infty} \frac{x^{\beta}}{(x^{2}+1)^{2}} \mathrm{~d}x$, consider the line integral of the complex function $f(z) = \frac{z^{\beta}}{(z^{2}+1)^{2}}$ on the complex plane. Here, $\beta$ is a real number and $0 < \beta < 1$. The closed integration path $C = C_{1} + C_{R} + C_{2} + C_{r}$ $(0 < r < 1 < R)$ is defined with semicircles and line segments as shown in Figure 3.1.
\begin{enumerate}
\item Using the residue theorem, calculate the line integral $\oint_{C} f(z) \mathrm{d}z$.
\item Express $\int_{C_{1}} f(z) \mathrm{d}z + \int_{C_{2}} f(z) \mathrm{d}z$ with the definite integral $\int_{r}^{R} \frac{x^{\beta}}{(x^{2}+1)^{2}} \mathrm{~d}x$.
\item Obtain $\lim_{R \rightarrow \infty} \int_{C_{R}} f(z) \mathrm{d}z$.
\item Obtain $\lim_{r \rightarrow 0} \int_{C_{r}} f(z) \mathrm{d}z$.
\item Using the previous results, calculate the definite integral $I$.
\end{enumerate}