Compute $\int_{0}^{\infty}\frac{x\sin 2x}{9+x^{2}} \, dx$

Before addressing the question regarding the limit of the integral over $\gamma_R$, it is important to first understand that

$$\int_0^\infty \frac{x\sin(2x)}{9+x^2}\,dx=\frac12\text{Im}\left(\int_{-\infty}^\infty \frac{xe^{i2x}}{9+x^2}\,dx\right)$$

Next, using the residue theorem we find that

$$\oint_\gamma \frac{ze^{i2z}}{9+z^2}\,dz=2\pi i \frac{3i(e^{-6})}{6i}=\pi ie^{-6}$$

Now, to address the question regarding the limit of the integral over $\gamma_R$, we first note that

$$\begin{align} \left|\int_0^\pi \frac{Re^{i\phi }e^{i2Re^{i\phi}}}{9+R^2e^{i2\phi}}\,iRe^{i\phi}\,d\phi\right|&\le \int_0^\pi \frac{R^2e^{-2R\sin(\phi)}}{|9+R^2e^{i2\phi}|}\\\\ &\le \frac{2R^2}{|R^2-9|}\int_0^{\pi/2} e^{-2R\sin(\phi)}\,d\phi\tag 1 \end{align}$$

Next, we use the fact that $\sin(\phi)\ge 2\phi/\pi$ for $\phi \in[0,\pi/2]$. Hence, we can write

$$\int_0^{\pi/2} e^{-2R\sin(\phi)}\,d\phi\le \int_0^{\pi/2} e^{-4R\phi/\pi}\,d\phi=\frac{1-e^{-2R}}{4R/\pi}\tag 2$$

Using $(2)$ in $(1)$ we find that

$$\begin{align} \left|\int_0^\pi \frac{Re^{i\phi }e^{i2Re^{i\phi}}}{9+R^2e^{i2\phi}}\,iRe^{i\phi}\,d\phi\right|\le \frac{2R^2}{|R^2-9|}\frac{1-e^{-2R}}{4R/\pi}\to 0\,\,\text{as}\,\,R\to \infty \end{align}$$

I'll use the duality of the convolution theorem to find the spectrum $\mathcal{F}\left(\frac{x}{9 + x^2}\right)$ as the convolution of the two spectra $\mathcal{F}\left(\frac{1}{9 + x^2}\right)$ and $\mathcal{F}\left(x\right)$. The spectrum of the first (Cauchy) function is well-known, being a symmetric mono-exponential, and the second spectrum is $\left(-i \sqrt{2\pi} δ'(ω)\right)$, being the differential operator in the Fourier domain times $i$.

$$\begin{align} I &= \int_{0}^{\infty}\frac{x\sin 2x}{9+x^{2}} \, dx \\ &=\frac{1}{2}\Im\left(\sqrt{2\pi}\mathcal{F}\left(\frac{x}{9 + x^2}\right)_{\omega = 2}\right) \\ &=\frac{1}{2}\Im\left(\frac{\sqrt{2\pi}\mathcal{F}\left(\frac{1}{9 + x^2}\right) \otimes \sqrt{2\pi}\mathcal{F}\left(x\right)}{2\pi}\right)_{\omega = 2} \\ &=\frac{1}{2}\Im\left(\frac 1 3 \sqrt{\frac\pi 2} e^{(-3|ω|)} \otimes -i \sqrt{2\pi} δ'(ω)\right)_{\omega = 2} \\ &=\frac{\pi}{2}\Im\left(i\left[\frac 1 3 e^{(-3|ω|)}\right]' \right)_{\omega = 2}\\ &=\frac{\pi}{2}\left[\frac 1 3 e^{(-3|ω|)}\right]'_{\omega = 2} \\ &=\frac{\pi}{2 e^{6}} \\ &\approx 0.003893... \end{align}$$

We may also use the Laplace transform to evaluate the integral. Using this useful identity of the Laplace transform we have $$I=\int_{0}^{\infty}\frac{x\sin\left(2x\right)}{x^{2}+9}dx\stackrel{2x\rightarrow x}{=}\int_{0}^{\infty}\frac{x\sin\left(x\right)}{x^{2}+36}dx=\int_{0}^{\infty}\frac{\cos\left(6x\right)}{x^{2}+1}dx.$$ Let us consider $$I\left(a\right)=\int_{0}^{\infty}\frac{\cos\left(ax\right)}{x^{2}+1}dx,\,a>0.$$ Then if we apply again the Laplace transform we have $$\mathfrak{L}\left(I\left(a\right)\right)\left(s\right)=\int_{0}^{\infty}\int_{0}^{\infty}\frac{\cos\left(ax\right)}{x^{2}+1}e^{-as}dadx=\int_{0}^{\infty}\frac{s}{\left(x^{2}+s^{2}\right)\left(1+x^{2}\right)}dx$$ hence using partial fractions $$\mathfrak{L}\left(I\left(a\right)\right)\left(s\right)=\frac{\pi}{2\left(s+1\right)}$$ so inverting $$I\left(a\right)=\color{red}{\frac{\pi}{2e^{a}}}.$$ Taking $a=6$ we have the result.