1. The problem is to evaluate the improper integral $$\int_0^\infty \frac{x}{(1+x^2)^2} \, dx$$ over the infinite interval from 0 to infinity. 2. We use the formula for integration by substitution and recognize that the integral involves a rational function with a quadratic denominator. 3. Let us use the substitution $u = 1 + x^2$, so that $du = 2x \, dx$ or $x \, dx = \frac{du}{2}$. 4. Changing the limits: when $x=0$, $u=1$; when $x \to \infty$, $u \to \infty$. 5. The integral becomes $$\int_1^\infty \frac{1}{2} \cdot \frac{1}{u^2} \, du = \frac{1}{2} \int_1^\infty u^{-2} \, du$$ 6. Integrate $u^{-2}$: $$\int u^{-2} \, du = -u^{-1} + C$$ 7. Evaluate the definite integral: $$\frac{1}{2} \left[-\frac{1}{u} \right]_1^\infty = \frac{1}{2} \left(0 + 1\right) = \frac{1}{2}$$ 8. Therefore, the value of the integral is $$\boxed{\frac{1}{2}}$$.