complex analysis - Help with this indefinite integral using residues?

Question: How to evaluate this integral using residues

$$\int_{0}^{\infty} \frac{x \sin x}{1 + x^2} dx$$

I integrate over the entire real axis and dividing it by 1/2 since the integrand is even, and then I do the thing with the turning it into

$$-\mathrm{Im}\bigg(\int f\cdot e^{ix} dx\bigg)$$ and integrate over the upper half plane by finding the residues there, and there's only one, at $i$.

So I put it into the equation $\frac{x e^{iz}}{2x}$ and get $\frac{1}{2e}$, so I get when plugging that in

$$\begin{align}\int_{0}^{\infty} \frac{x \sin x}{1 + x^2} dx &= .5 \int_{-\infty}^{\infty} \frac{x \sin x}{1 + x^2} dx \\&= .5 -\operatorname{Im} \Bigg[2\pi i \cdot \operatorname{Res}\bigg(\frac{x e^{ix}}{ 1 + x^2}; i\bigg)\Bigg]\\&= .5 \cdot \bigg[-\operatorname{Im}\Big(2\pi i\cdot \frac{1}{2e}\Big)\bigg] \\&= -\frac{\pi}{2e}\end{align}$$

But the answer's positive.

What did I mess up on?